WO2021115618A1

WO2021115618A1 - First and second communication devices for pilot-less transmissions in a communication system

Info

Publication number: WO2021115618A1
Application number: PCT/EP2019/085120
Authority: WO
Inventors: Yi Qin; Renaud-Alexandre PITAVAL
Original assignee: Huawei Technologies Co., Ltd.
Priority date: 2019-12-13
Filing date: 2019-12-13
Publication date: 2021-06-17

Abstract

The invention relates to first and second communication devices for pilot-less transmissions in a communication system. The first communication device obtains bits for transmission (B_c) and uses a linear mapping function (f) to a vector (B_g) for generating a constellation symbol based on the bits for transmission (B_c). Thereafter, a constellation symbol (x) is generated based on the vector (B_g) for generating a constellation symbol, wherein the generated constellation symbol (x) comprises multiple complex number elements. The constellation symbol (x) can be mapped onto time-frequency resources and transmitted to the second communication devices. Thereby, the average bit difference between neighbor constellation symbols is reduced and hence the bit error rate at demodulation can be reduced. Furthermore, the invention also relates to corresponding methods and a computer program.

Description

FIRST AND SECOND COMMUNICATION DEVICES FOR PILOT-LESS TRANSMISSIONS IN A COMMUNICATION SYSTEM

Technical Field

The invention relates to first and second communication devices for pilot-less transmissions in a communication system. Furthermore, the invention also relates to corresponding methods and a computer program.

Background

In many wireless communication systems, such as Long Term Evolution (LTE) and New Radio (NR) systems, data signals are usually transmitted with reference signals (RSs), also referred to as pilots or pilot signals, which enables channel estimation subsequently used for data detection at the receiver.

For example, in NR systems, data bits for transmission are first coded at the transmitter based on channel coding like Reed-Muller (RM) code, Low-Density Parity-Check (LDPC) code or Polar code, and then modulated to complex symbols by using binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), 16 quadrature amplitude modulation (16QAM), etc. After modulation, the modulated symbols are transmitted with RSs which are known at both the transmitter and the receiver. The receiver first estimates the radio channel based on the received RSs, and then demodulates the received data symbols based on the estimated radio channel. The receiver finally obtains the transmitted data bits after channel decoding from the output of the demodulator. Such kind of transmission is also known as pilot-based transmissions. In pilot-based transmission, some Time-Frequency (T-F) resources are dedicated to RS and thus do not convey any data bits.

Alternatively, pilot-less transmission has been proposed, which requires no RSs and no channel estimation at the receiver. Pilot-less transmissions can use all T-F resources for data transmission, and therefore may achieve better performance, e.g., lower block error rate (BLER) or higher throughput in the system. In particular, for pilot-less transmissions, the transmitter needs to jointly map the data bits for transmission (or coded bits) to multiple complex symbols. Since the output of the mapping is more than one complex symbol, it is referred to as multi-dimensional modulation for pilot-less transmission.

The following example shows how such kind of modulation works in which there is 1 antenna at transmitter and N ³ 1 antennas at receiver. There are K number of bits for modulation at the transmitter side. These K bits may be data bits, data bits with Cyclic Redundancy Check (CRC) bits, coded data bits or coded data and CRC bits. Pilot-less transmission can be described with 4 major steps:

1. Modulation: the transmitter maps the K bits to a row vector x = x_t with T complex symbols, where T > 1 is an integer, and x_t belongs to a set C of vectors of same length. The set C is called a multi-dimensional constellation and any vector in c_έ in C can be uniquely labelled by a vector of K bits. Moreover, we consider that all vectors in C are normalized.

2. Transmission: the transmitter maps these T complex symbols to T number of T-F resources, and then transmits them to a receiver over a radio channel.

3. Detection: the receiver receives N x T complex symbols, e.g. denoted as matrix Y, on N receiver antennas and T number of T-F resources, where N ³ 1. Then the receiver try to detect x_t by the following rule: the detected x_r is the vector in C that maximizes R(X) = xY^HYx^H among all x in C, where R(x) describes the correlation between x and Y and thus a large R(X_r) implies that the transmitted vector x_t is very likely to be x_r.

4. Demodulation: the receiver obtains the K bits from the detected vector x = x_r based on the one-to-one labelling of the vectors in C. The demodulated K bits are the same as the transmitted K bits only if x was correctly detected.

By the above example it is realized that channel estimation is not required and hence RSs not necessary and hence all T-F resources can be utilized for data transmission, which may provide higher resource utilization. The detection rule used in step 3 is valid only when the channel stays roughly the same on the T T-F resources. However, due to mobility and multipath channel, the channel on each T-F resource may vary significantly, especially for large T-F resources. One solution is to divide the total T-F resources into B number of smaller T-F resource blocks such the channel is rather constant within each block. In this case, multidimensional modulation can be applied to each T-F resource blocks. In order to achieve good BLER performance from the multiple T-F resource blocks, channel coding is necessary and should be applied among multiple T-F resource blocks. As mentioned, in LTE and NR, several channel coding methods are utilized, such as RM code, Polar code, LDPC code, and Turbo code. All of these channel codes are binary coding, and therefore modulation with good bit- error-rate (BER) performance is desired.

Summary

An objective of embodiments of the invention is to provide a solution which mitigates or solves the drawbacks and problems of conventional solutions. The above and further objectives are solved by the subject matter of the independent claims. Further advantageous embodiments of the invention can be found in the dependent claims.

According to a first aspect of the invention, the above mentioned and other objectives are achieved with a first communication device for a communication system, the first communication device being configured to obtain bits for transmission; obtain a vector for generating a constellation symbol using a linear mapping function based on the bits for transmission; and generate a constellation symbol based on the vector for generating a constellation symbol, wherein the generated constellation symbol comprises multiple complex number elements.

The bits for transmission may be the channel coded bits, e.g., output of LDPC, Turbo or Polar code. The bits for transmission may also be interleaved bits. The bits for transmission may also be the information bit, e.g., control information bit. The bits for transmission may also be the error check bits, e.g., CRC bits, or any other suitable bits for transmission.

An advantage of the first communication device according to the first aspect is that the average bit difference between neighbor constellation symbols is reduced. Therefore, the bit error rate at demodulation will be reduced. Moreover, if channel coding is applied, the block error rate of the code block will also be reduced at the receiver.

In an implementation form of a first communication device according to the first aspect, the linear mapping function is based on a full rank block diagonal matrix comprising at least two blocks, wherein one of the at least two blocks is an identity matrix, and wherein the rows or columns of at least one block of the full rank block diagonal matrix includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an all-zero vector.

An advantage with this implementation form is that full rank matrix guarantees the linear mapping is a one to one mapping. Different blocks in full rank block diagonal matrix on the other hand minimize bit difference between neighboring constellation symbols in this case.

In an implementation form of a first communication device according to the first aspect, the first communication device is configured to map the bits for transmission using the linear mapping function to obtain an intermediate vector; obtain a transformation vector having m complex number elements based on the intermediate vector, where p is a prime number and p^m is the length of the constellation symbol; and obtain the vector for generating a constellation symbol based on an XOR operation of the intermediate vector and the transformation vector.

In an example, for two bit vectors a and b, the XOR operation can be understood as a XOR b = (a + b)mod2, where mod2 is the modulo 2 operator.

An advantage with this implementation form is that by the transformation the full rank nonblock diagonal matrix can be used instead of a block diagonal matrix.

In an implementation form of a first communication device according to the first aspect, the linear mapping function is based on a full rank matrix, and wherein the rows or columns of the full rank matrix includes complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an all-zero vector.

An advantage with this implementation form is that the full non-block diagonal matrix can achieve smaller average bit difference between neighboring constellation symbols than block diagonal matrix in this case.

In an implementation form of a first communication device according to the first aspect, the length N of the generated constellation symbol is N = p^m, and wherein the linear mapping using the linear mapping function is within a Galois field GF(p), where p is a prime number and m is a positive integer.

An advantage with this implementation form is that it guarantees that the vector for generating a constellation symbol is in Galois field GF(p).

In an implementation form of a first communication device according to the first aspect, p > 2, and the first communication device is configured to map the bits for transmission to a Galois field vector, wherein the complex number elements of the Galois field vector belong to {0,1, ... ,p - 1}; and map the Galois field vector) using the linear mapping function to obtain the vector for generating a constellation symbol.

An advantage with this implementation form is that since the linear mapping is in GF(p ), mapping the bits for transmission to a Galois field vector can guarantee that most of the generation vector is mapped, and therefore that most constellation symbols are indexed.

In an implementation form of a first communication device according to the first aspect, the first communication device is configured to separate the bits for transmission to obtain t number of vectors, wherein t is a positive integer dependent on the length N of the constellation symbol; map the t number of vectors using the linear mapping function to obtain t number of symbols for generating a constellation symbol; and generate the constellation symbol based on the t number of symbols for generating a constellation symbol.

An advantage with this implementation form is that since for some constellation symbol length the linear mapping cannot be directly applied to the bits for transmission. So, the bits for transmission are separated into multiple vectors before linear mapping.

In an implementation form of a first communication device according to the first aspect, the constellation symbol is the Kroneck product of the t number of symbols for generating a constellation symbol.

In this implementation form the symbols generated by the separate vectors are combined to generate the constellation symbol.

In an implementation form of a first communication device according to the first aspect, the linear mapping function is based on a full rank block diagonal matrix G, and wherein the rows or columns of at least one block of the full rank block diagonal matrix G includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to the constellation symbol generated by an all-zero vector.

An advantage with this implementation form is that by using the full rank block diagonal matrix the average bit difference between neighboring symbols can be made as small as possible. In an implementation form of a first communication device according to the first aspect, the length N of the generated constellation symbol

where p_t are prime numbers and m _t are positive integers for i = 1,2 ... t, where t is a positive integer, and wherein the linear mapping using the linear mapping function for an i-th bit vector of the t number of vectors is within a Galois field GF(p_j), where p_t is a prime number and m _t is a positive integer.

An advantage with this implementation form is that for constellation symbol with length N =

the average bit difference between neighboring symbols can be made as small as possible.

In an implementation form of a first communication device according to the first aspect, p_t > 2, and the first communication device is configured to map the i-th bit vector of the t number of vectors to a Galois field vector, wherein the complex number elements of the Galois field vector belong to {0,1, ... , p_t - 1}; and map the Galois field vectors using the linear mapping function to obtain the vector for generating a constellation symbol.

An advantage with this implementation form is that as the linear mapping for each individual symbol is in GF(p). So, mapping the i-th bit vector of the t number of vectors to a Galois field vector, can assure that most of the generation vector B_g is mapped, and therefore most constellation symbols are indexed.

In an implementation form of a first communication device according to the first aspect, the first communication device is configured to map the generated constellation symbol onto time-frequency resources; and transmit the generated constellation symbol mapped onto the time-frequency resources

According to a second aspect of the invention, the above mentioned and other objectives are achieved with a second communication device for a communication system, the second communication device being configured to receive a symbol mapped onto time-frequency resources; demodulate the received symbol based on a symbol constellation, wherein the symbol constellation comprises multiple constellation symbols, each constellation symbol comprising multiple complex number elements and wherein each constellation symbol is generated based on a vector for generating a constellation symbol, wherein the vector for generating a constellation symbol is obtained by using a linear mapping function based on possible transmitted bits. An advantage of the second communication device according to the second aspect is that the average bit difference between neighbor constellation symbols is reduced. Therefore, the bit error rate at demodulation will be reduced. Moreover, if channel coding is applied, the block error rate of the code block will also be reduced at the receiver.

In an implementation form of a second communication device according to the second aspect, the linear mapping function is based on a full rank block diagonal matrix comprising at least two blocks, wherein one of the at least two blocks is an identity matrix, and wherein the rows or columns of at least one block of the full rank block diagonal matrix includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an all-zero vector.

In an implementation form of a second communication device according to the second aspect, the vector for generating a constellation symbol is based on an XOR operation of an intermediate vector and a transformation vector, wherein the transformation vector has m complex number elements based on the intermediate vector, where p is a prime number and P^m is the length of the constellation symbol, and wherein the intermediate vector is obtained by mapping possible transmitted bits using the linear mapping function.

In an implementation form of a second communication device according to the second aspect, the linear mapping function is based on a full rank matrix, and wherein the rows or columns of the full rank matrix includes complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an all-zero vector.

An advantage with this implementation form is that the full non-block diagonal matrix can achieve smaller average bit difference between neighboring constellation symbols than block diagonal matrix in this case. In an implementation form of a second communication device according to the second aspect, the length N of the constellation symbol is N = p^m, and wherein the linear mapping using the linear mapping function is within a Galois field GF(p ), where p is a prime number and m is a positive integer.

In an implementation form of a second communication device according to the second aspect, p > 2, and wherein the possible transmitted bits are mapped to a Galois field vector, wherein the complex number elements of the Galois field vector belong to {0,1, ... , p - 1], and wherein the vector for generating a constellation symbol is obtained by mapping the Galois field vector using the linear mapping function.

An advantage with this implementation form is that since the linear mapping is in GF(p ), mapping the bits for transmission to a Galois field vector can guarantee that most B_g are mapped, and therefore that most constellation symbols are indexed.

In an implementation form of a second communication device according to the second aspect, the constellation symbol is based on t number of symbols for generating a constellation symbol, where t is a positive integer dependent on the length N of the constellation symbol , and wherein the t number of symbols for generating a constellation symbol is obtained by mapping t number of vectors using the linear mapping function, wherein the t number of vectors is a separation of the possible transmitted bits.

In an implementation form of a second communication device according to the second aspect, the constellation symbol is the Kroneck product of the t number of symbols for generating a constellation symbol.

In this implementation form the symbols generated by the separate vectors are combined to generate the constellation symbol. In an implementation form of a second communication device according to the second aspect, the linear mapping function is based on a full rank block diagonal matrix G, and wherein the rows or columns of at least one block of the full rank block diagonal matrix G includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to the constellation symbol generated by an all-zero vector.

An advantage with this implementation form is that by using the full rank block diagonal matrix the average bit difference between neighboring symbols can be made as small as possible.

In an implementation form of a second communication device according to the second aspect, the length N of the constellation symbol is N =

r_έ are prime numbers and m_t are positive integers for i = 1,2 ... t, where t is a positive integer, and wherein the linear mapping using the linear mapping function for an i-th bit vector of the t number of vectors is within a Galois field GF(pi), where p_t is a prime number and

is a positive integer.

An advantage with this implementation form is that for constellation symbol with length N = p™¹ p™² ...p_t ^mt the average bit difference between neighboring symbols can be made as small as possible.

In an implementation form of a second communication device according to the second aspect, Pi > 2, and wherein the vector for generating a constellation symbol is obtained by mapping Galois field vectors using the linear mapping function, wherein the Galois field vectors is obtained by mapping the i-th bit vector of the t number of vectors to the Galois field vector, wherein the elements of the Galois field vector belong to {0,1, ... , p_t - 1}.

An advantage with this implementation form is that as the linear mapping for each individual symbol is in GF(p). So, mapping the i-th bit vector of the t number of vectors to a Galois field vector, can guarantee most of the generation vector B_g is mapped, and therefore most constellation symbols are indexed.

According to a third aspect of the invention, the above mentioned and other objectives are achieved with a method for a first communication device, the method comprises obtaining bits for transmission; obtaining a vector for generating a constellation symbol using a linear mapping function based on the bits for transmission; and generating a constellation symbol based on the vector for generating a constellation symbol, wherein the generated constellation symbol comprises multiple complex number elements.

The method according to the third aspect can be extended into implementation forms corresponding to the implementation forms of the first communication device according to the first aspect. Hence, an implementation form of the method comprises the feature(s) of the corresponding implementation form of the first communication device.

The advantages of the methods according to the third aspect are the same as those for the corresponding implementation forms of the first communication device according to the first aspect.

According to a fourth aspect of the invention, the above mentioned and other objectives are achieved with a method for a second communication device, the method comprises receiving a symbol mapped onto time-frequency resources; demodulating the received symbol based on a symbol constellation, wherein the symbol constellation comprises multiple constellation symbols, each constellation symbol comprising multiple complex number elements and wherein each constellation symbol is generated based on a vector for generating a constellation symbol, wherein the vector for generating a constellation symbol is obtained by using a linear mapping function based on possible transmitted bits.

The method according to the fourth aspect can be extended into implementation forms corresponding to the implementation forms of the second communication device according to the second aspect. Hence, an implementation form of the method comprises the feature(s) of the corresponding implementation form of the second communication device.

The advantages of the methods according to the fourth aspect are the same as those for the corresponding implementation forms of the second communication device according to the second aspect.

The invention also relates to a computer program, characterized in program code, which when run by at least one processor causes said at least one processor to execute any method according to embodiments of the invention. Further, the invention also relates to a computer program product comprising a computer readable medium and said mentioned computer program, wherein said computer program is included in the computer readable medium, and comprises of one or more from the group: ROM (Read-Only Memory), PROM (Programmable ROM), EPROM (Erasable PROM), Flash memory, EEPROM (Electrically EPROM) and hard disk drive.

Further applications and advantages of the embodiments of the invention will be apparent from the following detailed description.

Brief Description of the Drawings

The appended drawings are intended to clarify and explain different embodiments of the invention, in which:

- Fig. 1 shows a first communication device according to an embodiment of the invention;

- Fig. 2 shows a method for a first communication device according to an embodiment of the invention;

- Fig. 3 shows a second communication device according to an embodiment of the invention;

- Fig. 4 shows a method for a second communication device according to an embodiment of the invention;

- Fig. 5 shows a wireless communication system according to an embodiment of the invention;

- Fig. 6 illustrates the structure of a constellation based on 2^nd order RM code;

- Fig. 7 shows a flow chart of a method according to an embodiment of the invention;

- Fig. 8 shows a flow chart of a method according to an embodiment of the invention;

- Fig. 9 shows a flow chart of a method according to an embodiment of the invention;

- Fig. 10 shows a flow chart of a method according to an embodiment of the invention;

- Fig. 11 shows a flow chart of a method according to an embodiment of the invention; and

- Fig. 12 and 13 show performance results of embodiments of the invention.

Detailed Description

With reference to the previous discussion about pilot-less transmission in NR a well-designed constellation for multi-dimensional modulation, e.g., constellation based on second-order RM code, can achieve some symbol-error-rate (SER). However, it does not necessarily imply a good BER performance if the labeling of each constellation point is not carefully designed. Therefore, how to map bits to constellation point in a given constellation, i.e., labelling method, becomes a very important when applying multi-dimensional modulation. Labelling can in this disclosure also be understood as a method of indexing for mapping bits to a constellation point in a given set of constellation points. If e.g. the label is a bit vector the labelling can be denoted as a binary index.

Labelling of a constellation may be defined as a one-to-one mapping from all K- bit vectors to all constellation points in a constellation. For multi-dimensional modulation, each constellation point is a vector of complex symbols with length T. Based on this labelling, for any K bit input, C, the output of T complex symbols x can be found. The bit vector may be called label or alternatively index of the corresponding complex symbol vector. The constellation point may be called constellation symbol or modulation symbol.

It has been recognized by the inventors that according to conventional solutions, the mapping rule is to guarantee that constellation point pair with large chordal distance is mapped to bit vector pair with large Hamming distance. The chordal distance of two normalized complex symbol vectors x₁ and x₂ is defined as d = l -

and the Hamming distance is defined as the number of bit difference between two bit vectors.

In particular, the following steps are used to generate labels for each constellation point:

• Generate Hamming distance matrix H of all N-bit vectors. Each element in H is the hamming distance between its row and column binary index. For example, if N = 2, H is a 4 x 4 matrix. The four binary indexes are 00,01 ,10,11 , and H is as follow

• Generate chordal distance matrix

the constellation matrix and each row vector of C_x (denoted as C_x(i for the i-th row) is a constellation point in the constellation.

• Generate the rank matrix R by marking the elements of each row of E in ascending order of chordal distance, and guarantee marking with the same number of 0, 1 , 2, ... as in H.

• Target: to find the optimal order of row of C_x that minimizes ||H - R|| .

From the target of conventional solutions as stated above, mentioned conventional solutions tries to match all constellation point pairs to the corresponding label pairs in the same order. If || H — R||_p = 0, it means that the chordal distance perfectly matches the Hamming distance. However, it is impossible to find the optimal solution of the order. So, sub-optimal solution is desired. Some searching algorithms, e.g., greedy searching, can be used to find good solution of C_x. There are at least three main drawbacks of conventional solutions identified by the inventors. According to the rule of minimizing the expression ||H - R|| , the conventional solutions try to match all elements in H to R with the same priority. However, if misdetection happens, the probability that one constellation point c_έ is misdetected as another point x_; is related to their chordal distance. Therefore, from the perspective of error rate, the priority of different constellation point pairs may be not the same, i.e., it is not optimal to match all elements in H to R with the same priority. Usually, H and R cannot be perfectly matched, and the assumption of same priority may lead to mismatch for some key constellation point pairs, where the key constellation point pair is defined as the pair mainly determines the error probability. The sub- optimal C_x is obtained by greedy searching, but the complexity is unacceptable for large constellation size. The constellation size may be a quantity of constellation points. Due to computer searching, there is only numerical solution.

Therefore, the inventors aim, among other things, to design a indexing/labelling solution to minimize BER for a given set of constellation points. For the first drawback, it can be found that the key constellation point pair is the one with minimum chordal distance. Hence, only match constellation point pairs with minimum chordal distance (neighbors) to label pairs with as minimum Hamming distance as possible is advantage. Similar idea is used in Gray labelling method for one-dimensional modulation, which requires the labels of neighbors has only one bit difference. However, for multi-dimensional modulation, there are usually too many neighbors. Since the number of IV-bit vectors that have only one bit difference from a given vector is N, it becomes impossible to apply Gray labelling if there are more than N neighbors for each constellation point. So, a design target is to minimize the average bit difference (or average Hamming distance) of labels for constellation point pairs with minimum chordal distance. To avoid the second and third drawbacks of conventional solutions, a structured labelling method is designed, and therefore searching is not needed. In order to design an effective structured labelling method, a structured constellation may be needed. Therefore, the constellation may be generated based on second order Reed-Muller code, which can achieve optimal SER performance in some cases. For example, for the case of vector length equals to 2^m and the constellation size is multiple of m bits, the constellation generated based on second order Reed-Muller code can achieve optimal minimum distance.

Fig. 1 shows a first communication device 100 according to an embodiment of the invention. In the embodiment shown in Fig. 1 , the first communication device 100 comprises a processor 102, a transceiver 104 and a memory 106. The processor 102 may be coupled to the transceiver 104 and the memory 106 by communication means 108 known in the art. The first communication device 100 may further comprise an antenna or antenna array 110 coupled to the transceiver 104, which means that the first communication device 100 may be configured for wireless communications in a wireless communication system.

That the first communication device 100 may be configured to perform certain actions can in this disclosure be understood to mean that the first communication device 100 comprises suitable means, such as e.g. the processor 102 and the transceiver 104, configured to perform said actions.

The processor 102 of the first communication device 100 may be referred to as one or more general-purpose central processing units (CPUs), one or more digital signal processors (DSPs), one or more application-specific integrated circuits (ASICs), one or more field programmable gate arrays (FPGAs), one or more programmable logic devices, one or more discrete gates, one or more transistor logic devices, one or more discrete hardware components, and one or more chipsets.

The memory 106 of the first communication device 100 may be a read-only memory, a random access memory, or a non-volatile random access memory (NVRAM).

The transceiver 104 of the first communication device 100 may be a transceiver circuit, a power controller, an antenna, or an interface which communicates with other modules or devices.

In embodiments, the transceiver 104 of the first communication device 100 may be a separate chipset or being integrated with the processor 102 in one chipset. While in some embodiments, the processor 102, the transceiver 104, and the memory 106 of the first communication device 100 are integrated in one chipset.

According to embodiments of the invention the first communication device 100 is configured to obtain bits for transmission B_c. The first communication device 100 is configured to obtain a vector B_g for generating a constellation symbol using a linear mapping function / based on the bits for transmission B_c. The first communication device 100 is configured to generate a constellation symbol x based on the vector B_g for generating a constellation symbol. The generated constellation symbol x comprises multiple complex number elements.

Fig. 2 shows a flow chart of a corresponding method 200 which may be executed in a first communication device 100, such as the one shown in Fig. 1. The method 200 comprises obtaining 202 bits for transmission B_c. The method 200 comprises obtaining 204 a vector B_g for generating a constellation symbol using a linear mapping function / based on the bits for transmission B_c. The method 200 comprises generating 206 a constellation symbol x based on the vector B_g for generating a constellation symbol. The generated constellation symbol x comprises multiple complex number elements.

Fig. 3 shows a second communication device 300 according to an embodiment of the invention. In the embodiment shown in Fig. 3, the second communication device 300 comprises a processor 302, a transceiver 304 and a memory 306. The processor 302 is coupled to the transceiver 304 and the memory 306 by communication means 308 known in the art. The second communication device 300 may be configured for wireless communications. The wireless communication capability is provided with an antenna or antenna array 310 coupled to the transceiver 304.

That the second communication device 300 is configured to perform certain actions can in this disclosure be understood to mean that the second communication device 300 comprises suitable means, such as e.g. the processor 302 and the transceiver 304, configured to perform said actions.

The processor 302 of the second communication device 300 may be referred to as one or more general-purpose CPUs, one or more DSPs, one or more ASICs, one or more FPGAs, one or more programmable logic devices, one or more discrete gates, one or more transistor logic devices, one or more discrete hardware components, and one or more chipsets.

The memory 306 of the second communication device 300 may be a read-only memory, a random access memory, or a NVRAM.

The transceiver 304 of the second communication device 300 may be a transceiver circuit, a power controller, an antenna, or an interface which communicates with other modules or devices.

In embodiments, the transceiver 304 of the second communication device 300 may be a separate chipset or being integrated with the processor 302 in one chipset. While in some embodiments, the processor 302, the transceiver 304, and the memory 306 of the second communication device 300 are integrated in one chipset.

According to embodiments of the invention the second communication device 300 is configured to receive a symbol Y mapped onto time-frequency resources, e.g. over a radio channel. The second communication device 300 is configured to demodulate the received symbol Y based on a symbol constellation. The symbol constellation comprises multiple constellation symbols, and each constellation symbol x comprises multiple complex number elements. Further, each constellation symbol x is generated based on a vector B_g for generating a constellation symbol, wherein the vector B_g for generating a constellation symbol is obtained by using a linear mapping function / based on possible transmitted bits B_c.

Fig. 4 shows a flow chart of a corresponding method 400 which may be executed in a second communication device 300, such as the one shown in Fig. 3. The method 400 comprises receiving 402 a symbol Y mapped onto time-frequency resources, e.g. over a radio channel. The method 400 comprises demodulating 404 the received symbol Y based on a symbol constellation, wherein the symbol constellation comprises multiple constellation symbols, and each constellation symbol x comprises multiple complex number elements. Further, each constellation symbol x is generated based on a vector B_g for generating a constellation symbol, wherein the vector B_g for generating a constellation symbol is obtained by using a linear mapping function / based on possible transmitted bits B_c.

Embodiments of the invention provide solutions for mapping constellation points to labels and to minimize the average bit difference (or average Hamming distance) of labels for constellation point pairs with minimum chordal distance by using linear mapping. The linear mapping function / can be expressed in many different ways. In the following disclosure nonlimiting examples and embodiments are presented.

According to embodiments of the invention three main methods and their variants denoted as method 1 , method 2 and method 3 are presented.

In method 1 / is a linear function as previously mentioned, and in an embodiment the linear mapping function / is based on a full rank block diagonal matrix G comprising at least two blocks. One of the at least two blocks is an identity matrix, and the rows or columns of at least one block of the full rank block diagonal matrix G includes part of complex number elements of vectors for generating constellation symbols which are closest to a constellation symbol generated by an all-zero vector.

In method 2, the first communication device 100 is configured to map the bits for transmission B_c using the linear mapping function / to obtain an intermediate vector B₅; obtain a transformation vector having m complex number elements based on the intermediate vector B₅, where p is a prime number and p^m is the length of the constellation symbol x; and obtain the vector B_g for generating a constellation symbol based on an XOR operation of the intermediate vector B_g and the transformation vector b_Bfl.

In method 3, the first communication device 100 is configured to: separate the bits for transmission B_c to obtain t number of vectors, wherein t is a positive integer dependent on the length N of the constellation symbol x; map the t number of vectors using the linear mapping function / to obtain t number of symbols x for generating a constellation symbol; and generate the constellation symbol x based on the t number of symbols x for generating a constellation symbol.

In embodiments of method 3, the constellation symbol x is the Kroneck product of the t number of symbols x for generating a constellation symbol.

When generating the constellation symbols, the first communication device may transmit the constellation symbols directly or after some processing, such as up conversion or power amplification.

Fig. 5 shows a communication system 500 according to an embodiment of the invention. The communication system 500 comprises a first communication device 100 and a second communication device 300 configured to operate in the communication system 500. For simplicity, the communication system 500 shown in Fig. 5 only comprises one first communication device 100 and one second communication device 300. However, the communication system 500 may comprise any number of first communication devices 100 and any number of second communication devices 300 without deviating from the scope of the invention.

In the communication system 500, the first communication device 100 communicates with the second communication device 300, and vice versa. The communication between the first communication device 100 and the second communication device 300 may e.g. be in the uplink (UL) and/or in the downlink (DL) via the Uu interface e.g. depending on if the first communication device 100 act as a network access node and the second communication device 300 as a client device or vice versa. Also, sidelink communication using sidelink interface between the first communication device 100 and the second communication device 300 is possible. In this particular non-limiting example, the first communication device 100 acts as a network node such as a gNB and the second communication device 100 act as a client device such as a UE. However, the revers case is obviously possible. It is illustrated in Fig. 5 how the second communication device 300 receives symbol Y over a radio channel in the DL. The communication system 500 in this disclosure includes but is not limited to: LTE, 5G or future wireless communication system.

A client device in this disclosure includes but is not limited to: a UE such as a smart phone, a cellular phone, a cordless phone, a session initiation protocol (SIP) phone, a wireless local loop (WLL) station, a personal digital assistant (PDA), a handheld device having a wireless communication function, a computing device or another processing device connected to a wireless modem, an in-vehicle device, a wearable device, an integrated access and backhaul node (IAB) such as mobile car or equipment installed in a car, a drone, a device-to-device (D2D) device, a wireless camera, a mobile station, an access terminal, an user unit, a wireless communication device, a station of wireless local access network (WLAN), a wireless enabled tablet computer, a laptop-embedded equipment, an universal serial bus (USB) dongle, a wireless customer-premises equipment (CPE), and/or a chipset. In an Internet of things (IOT) scenario, the client device may represent a machine or another device or chipset which performs communication with another wireless device and/or a network equipment.

The UE may further be referred to as a mobile telephone, a cellular telephone, a computer tablet or laptop with wireless capability. The UE in this context may e.g. be portable, pocket- storable, hand-held, computer-comprised, or vehicle-mounted mobile device, enabled to communicate voice and/or data, via the radio access network, with another entity, such as another receiver or a server. The UE can be a station (STA), which is any device that contains an IEEE 802.11 -conformant media access control (MAC) and physical layer (PHY) interface to the wireless medium (WM). The UE may also be configured for communication in 3GPP related LTE and LTE-Advanced, in WiMAX and its evolution, and in fifth generation wireless technologies, such as NR.

The network access node in this disclosure includes but is not limited to: a NodeB in wideband code division multiple access (WCDMA) system, an evolutional Node B (eNB) or an evolved NodeB (eNodeB) in LTE systems, or a relay node or an access point, or an in-vehicle device, a wearable device, or a gNB in the fifth generation (5G) networks.

Further, the network access node herein may be denoted as a radio network access node, an access network access node, an access point, or a base station, e.g. a radio base station (RBS), which in some networks may be referred to as transmitter, “gNB”, “gNodeB”, “eNB”, “eNodeB”, “NodeB” or “B node”, depending on the technology and terminology used. The radio network access nodes may be of different classes such as e.g. macro eNodeB, home eNodeB or pico base station, based on transmission power and thereby also cell size. The radio network access node can be a station (ST A), which is any device that contains an IEEE 802.11 -conformant MAC and PHY interface to the wireless medium. The radio network access node may also be a base station corresponding to the 5G wireless systems.

For providing even deeper understanding of the invention, further non-limiting embodiments will be described in the following disclosure. Generally, variants of the three methods 1 , 2 and 3 will be presented.

Expression of constellation point (CP)

Firstly, however it is herein given an example of how Reed-Muller (RM) code can be used to design a set of constellation points used herein. The second order Reed-Muller code is designed for the constellation with vector length 2^m , i.e., each constellation point in the constellation is a vector with length 2^m. The constellation size is 2^(r+2)m, where r is

an integer. In this case, the i-th constellation point can be expressed as

where: l is the binary vector representation of L - 1 , where L = 1,2, ...2^m is the index of element in c_έ.

For example, if the vector length T = 2² (i.e., m = 2 ) and L = 3 , then l = [0, 1]^T , and l(j ) 2^;_1 = 2 = L - 1, fc® is the bit vector of the first m bits of B_g, where B_g is the binary vector representation of i - 1. B_g is called the generation vector of constellation based on 2^nd order RM code. For example, if there are N = 2^s constellation points in the constellation and i = 10, then B_g = [1, 0, 0, 1, 0 ], and fc® = [1, 0]. P® is a binary symmetric matrix of size m x m selected from a set of binary matrices DG(m,r ) and r £

The set DG(m,r ) is the

Delsart-Goethals (DG) set. DG(m,r ) is a (r + l)m-dirnentional binary subspace leading to a collection of 2^(r+1)m symmetric matrices such that any nonzero matrices in D G(m,r) has rank at least equal to R = m - 2r. In particular, P® can be expressed as P® =

P® mod2, where P® is a symmetric binary matrix of size m x m with rank m - u.

For each u, P® is generated by m basis matrices

where b_t is the t-th bit in B_g. Therefore, the second m bits of B_g determine P₀ ^W, and P^ is determined by the (u + 2)-th m bits of B_g.

For the case that m is even, the maximum constellation size is

when r = = y -

1. However, the corresponding DG set DG (m, [^y^]) does not include all symmetric matrices so that one can construct a larger set DG (m, py^j) in which all non-zero matrices have rank at least equal to R = 1. This last set has only y extra dimensions over DG (m Therefore,

the constellation size can be extended to 2\^~'^m .

can be expressed as P® =

and where the Q₁ Q₂, ... , Q_m/2 are the extra dimensions.

Structure of constellation

The structure of the constellation based on 2^nd order RM code is illustrated in Fig. 6. On the right-hand side in Fig. 6, the structure of the constellation is as follow:

• The constellation is a set of 2^(r+2)m CPs.

• The constellation includes 2^(r+1^^m set of CPs, and the CPs in each set are orthogonal to each other. So it is called orthogonal CP set or level 0 CP set. Each level 0 CP set includes 2^m CPs.

• Every 2^m level 0 CP sets compose one level 1 CP set, and the CPs in the same level

1 CP set have maximum correlation 2^_m/2. The correlation of two CPs C_έ and X_; is defined as |x_tx |.

• Every 2^m level 1 CP sets compose one level 2 CP set, and the CPs in the same level

2 CP set have maximum correlation 2

.

• 2^m level r CP sets compose one level r + l CP set, and the CPs in the level r + 1

CP set have maximum correlation 2^r~ . The level r + 1 CP set is the whole constellation if the constellation is not extended.

If m is even, and the constellation size is extended to

there are 2^m/2 level r + 1 CP set in the constellation, and the maximum correlation of CPs in the constellation

On the left-hand side in Fig. 6, the relation between generation vector B_g and constellation point is shown where:

• The first m bits of B_g determine the index of the CP within its level 0 CP set.

• The second m bits of B_g determines the index of its level 1 CP set within its level 0 CP set.

•

• The (r + 2)-th m bits of B_g determine the index of its level r CP set within its level r + 1 CP set or the constellation if it is not extended.

• If m is even, and the constellation size is extended to

the last — 2 bits of B 9„ determine the index of its level r + 1 CP set.

It should be noted that the order of bits in B_g may changes in some applications, but the index of one level of CP set is still determined by m bits in B_g.

Example of constellation based on 2^nd order RM code In this non-limiting example, the vector length of CP is 2² = 4, which means m = 2. The codebook size is 2⁴ = 16, which means r = 0.

The basic matrices Q^⁰⁾, Qj,⁰⁾ can be found as:

Therefore, according to the generation function in Eq. (1) and (2), the following CPs can be obtained according to Table 1 .

Table 1

It can be found that only the first m = 2 bits are different among the first 4 CPs. Thus, the first four CPs compose a level 0 CP set. Similarly, there are 4 level 0 CP sets: (0,1 , 2, 3), (4, 5, 6, 7), (8,9,10,11), (12,13,14,15), and the CPs within each level 0 CP set are orthogonal. The constellation is a level 1 CP set, which includes 2^m = 4 level 0 CP set, and the correlation between CPs within it is no larger that Since the m = 2 is even, the constellation can be extended to 2\^~)^m = 32. The extra basic matrix is:

«?^*=[; it

The CPs in the constellation are given by Table 2. The maximum correlation of CPs in the constellation is V2=.

Table 2

Labelling design for constellation generated based on second order Reed-Muller code with vector length of constellation point 2^m. In this case, the constellation and generation vector B_g introduced previously are used. The codebook size is 2^(r+2)m where r = -1, 0, 1, and

r may also equal to if m is even. For r = -1, the codebook is constructed by taking any orthogonal subset of size 2^m from the codebook with r = 0.

Method 1 The labelling according to method 1 may include the following major steps where the corresponding flow chart is given in Fig. 7.

Step 1 in Fig. 7, the first communication device 100 determines the bits as a bit vector B_c for modulation. Step 2 in Fig. 7, the first communication device 100 determines the bits as a bit vector B_g for constellation generation by a linear mapping from B_c using a linear mapping function /, which can be written as B_g = (B_cG)mod2.

Step 2a in Fig. 7, the bit vector B_c = [0,0, ... ,0] is mapped to generation vector

= [0,0, ... ,0] .

Step 2b in Fig. 7, sort CPs based on the chordal distance to CP with generation vector

= [0,0, ...,0] in ascending order. The corresponding generation vectors are recorded as

Step 2c in Fig. 7, G is a binary matrix of size (r + 2 )m x (r + 2 )m, which is expressed as:

where I_m is an m x m identity matrix, and g_t are the last (r + l)m bits in the vector B_g ^ki) where the subset of CP generation vectors , k_t > 0, is selected such that it

minimizes the sum chordal distance from the CP with label

while at the same time guarantees that G is full rank, i.e., the rank of G is (r + 2 )m.

Step 3 in Fig. 7, the first communication device 100 determines the constellation point as symbol vector x based on B_g according to Eq. (1) and (2).

With reference to the flow chart in Fig. 7 once more the following more detailed steps are performed.

Step 1 in Fig. 7, the bits for transmission B_c are obtained as input. The bits for transmission B_c may be the channel coded bits, e.g., output of LDPC, Turbo or Polar code. The bits for transmission B_c may also be interleaved bits. The bits for transmission B_c may also be the information bit, e.g., control information bit. The bits for transmission B_c may also be the error check bits e.g., CRC bits.

Step 2 in Fig. 7, B_g is compute according to B_g = (B_cG)mod2. Since for any input bits B_c the same block diagonal matrix G can be used. G can be obtained before the transmission or before step 1. There are three options how G is obtained, namely, • Option 1 : G is determined based on the CPs in the constellation according to the following steps 2a to 2d. The CPs in the constellation can be determined by the constellation size and vector length of the CP. The constellation size and vector length of the CP can be configured by the network of the communication system, by separate configuration or jointly coded configuration.

• Option 2: G is determined based on the CPs in the constellation, according to a predefined mapping between them. The CPs in the constellation can be determined in the same way as in option 1 .

• Option 3: G is configured by the network of the communication system. For example, the network sends the configuration to the first communication device 100 to select one G from the pre-defined set of matrices. The pre-defined set of matrices can be defined in specification or determined based on the constellation.

Step 2a in Fig. 7, the initial mapping results in simple expression of the quasi-Gray labelling, i.e. mapping

Step 2b in Fig. 7, the sorting of CPs guarantees the chordal distance cL© ₍₀₎ £

Since there may be some CPs with the same distance to

any order between these CPs is feasible.

Step 2c in Fig. 7, matrix G is generated and by using this structure of G can guarantee small bit difference for at least (r + 1 )m neighbors for each CP. The matrix I_m can be any full rank matrix of size m x m. Any order of the last (r + 1 )m rows of G is feasible. Moreover, in step 2c we do not need all sorted

therefore as an option for step 2c we can list the first (r + 1 )m closest CP to

guarantee that G is full rank. The following sub-steps are to generate G as above in details

• Sub-Step 2c- 1) Set G₀ = [I-m 0 mx(r+l)m ], i = 0

• Sub-step 2c-2) Generate temporary matrix

• Sub-step 2c-3) If G_temp is full rank, then G_i+1 = G_temp, otherwise G_i+1 = G_;

• Sub-step 2c-4) if the rank of G_i+1 is not (r + 2 )m, go to sub-step 2c-5), otherwise, go to sub-step 2c-6).

• Sub-step 2c-5) i = i + 1, and go to sub-step 2c-2).

• Sub-step 2c-6) output G = G_i+1. Steps 2a to 2c can in other words be formulated as that the linear mapping function / is based on a full rank block diagonal matrix G, and wherein the rows or columns of at least one block of the full rank block diagonal matrix G includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to the constellation symbol generated by an all-zero vector.

Step 2, 2a, 2b, and 2c in Fig. 7 can be extended by using initial mapping from arbitrary bit vector B‘^nit to arbitrary generation vector B ^u. In that case, step 2, 2a, 2b, and 2c are different from them in Fig.7 and there is an additional step of 2d following 2c, which is not shown in Fig.7

Step 2 becomes: the first communication device 100 determines the bits as a bit vector B _g for constellation generation by a mapping from B_c, which can be written as B_g = (B_CG + B₀)mod2 or B^ = (B_CG — B₀)mod2.

Step 2a becomes, the bit vector B‘^nit is mapped to generation vector B™^ic.

Step 2b becomes, sort CPs based on the chordal distance to CP with generation vector tna _jn ascending order. The corresponding generation vectors are recorded as

Step 2c becomes, G is a binary matrix of size (r + 2)m x (r + 2)m , which can be expressed as:

where I_m is an m x m identity matrix, g_init is the last (r + 1 )m bit of B™^ic, and the set is selected as previously.

Step 2d (not shown in Fig. 7), B₀ = (B‘^nitG + B"^lit)mod2 is a constant.

Step 3 in Fig. 7, the first communication device 100 determines the constellation point (as symbol vector x = [x₁,x₂, ... ,x₂ ^m ]) based on B_g according to Eq. (1) and (2). Another option is that there is a predefined table about the mapping between generation vectors and symbol vectors or CPs. In this case, the symbol vector x is determined by B_g according to the mapping from such a predefined table. It should be noticed that the symbol vector x may be not the symbol mapped to T-F resource. For example, the symbols mapped to time-frequency resource may be [c₁x₁, c₂x₂, ... ,c_2mx_2m], where c_t are constant and fixed for any x. Example of Method 1

In this example, the vector length of CP is 2² = 4, which means m 2. The constellation size is 2^s 32. The bit vector [0,0, 0,0,0] is mapped to generation vector [0,0, 0,0,0] Since r = 1/2, only (r + l)m = 3 generation vectors of neighbors from [0,0, 0,0,0] are needed. The neighbors of CP with generation vector [0,0, 0,0,0] are listed as follows in Table 3.

Table 3

In order to generate G, three other CPs are needed. In this example, we select the 1^st, 3^rd and 5^th CPs. According to step 2-c, the matrix G can be obtained as:

-1 0 0 0 0-

0 1 0 0 0

G = 0 0 1 0 1

0 0 0 1 1

-0 0 1 1 1-

Then, for a given bit vector B_c, the corresponding generation vector can be determined by B (B_cG)mod2. The results are listed as follow in Table 4.

Table 4

If comparing the proposed indexing/labelling with generation vector based on indexing/labelling and random indexing/labelling, the average bit difference in average Hamming distance from the different indexing methods for constellation point pairs with minimum chordal distance is 1.667, 3, and slightly more than 3, respectively. It can be found that the proposed solution significantly decreases the average bit difference in average Hamming distance of labels for neighbors. For this small example, a random search was performed over all possible indexing/labelling and we observe that the proposed method provides the best minimum average Hamming distance among the neighbors. Method 2

Method 2 is based on the method 1 described above. A basic difference is that an intermediate variable B_g is first determined based on B_c according to a similar linear mapping as in method 1 ; step 2a and 2c are however different. After that, a non-linear mapping is used to determine

The indexing according to method 2 may include the following steps and the corresponding flow chart is shown in Fig. 8.

Step 1 in Fig. 8, the first communication device 100 obtains a bit vector B_c for modulation.

Step 2 in Fig. 8, the first communication device 100 computes the intermediate bits as a bit vector B _g by a linear mapping from B_c using linear mapping function /, which can be written as B_g = (B_cG)mod2.

Step 2a in Fig. 8, B_g = B₅® [ b_Bg , li_X(r+i)m] is generated where B_g is the generation vector. The one-to-one mapping between B_g and B_g satisfies the following property: for two arbitrary CPs, i.e. X_j and x_; with generation vectors B^ and B^, their intermediate bit vector are B®

, respectively. Given one neighbor of c_έ corresponds to intermediate bit vector t_{here muS}† _{be a nej}ghbor of _Xj corresponding to the intermediate bit vector

gO-neighbor) _satisfy;_ng.

One possible way to get B_g from B_g is that the last (r + 1 )m bits of B_g and corresponding B_g are the same; and the first m bits satisfy

is a bit vector of length m determined by the last (r + 1 )m bits of B_g or B_g.

Step 2b in Fig. 8, the bit vector B_c = [0,0, ... ,0] is mapped to

= [0,0, ... ,0] .

Step 2c in Fig. 8, CPs are sorted based on the chordal distance to CP with

= [0,0, ... ,0] in ascending order. The corresponding intermediate vectors are recorded as

Step 2d in Fig. 8, G is a binary matrix of size (r + 2 )m x (r + 2 )m, which can be expressed as:

where the subset of CP generation vector

selected such that it minimizes the sum chordal distance from the CP with label

while at the same time guarantees that G is full rank, i.e., the rank of G is (r + 2)m.

Step 3 in Fig. 8, the first communication device 100 determines B_g according to B_g based on the predefined mapping between them, i.e. B_g = B₅® [b_Sg, 0_1X(r+1)m]

Step 4 in Fig. 8, the first communication device 100 computes the constellation point as symbol vector x based on B_g according to Eq. (1) and (2) in method 1. Thereafter, the symbol vector x is outputted.

With reference to the flow chart in Fig. 8 the following more detailed steps can be performed. Step 1 in Fig. 8 is the same as step 1 in Fig. 7.

Step 2 in Fig. 8 is the same as step 2 in Fig. 7.

Step 2a in Fig. 8, B_g is generated and any mapping between B_g and B_g satisfying the property is feasible. There may be a predefined table indicating the mapping between B_g and B₅, or there may be formula to obtain B_g based on B_g.

Step 2b in Fig. 8, is the same as step 2b in Fig. 7, this initial mapping results in simple expression of the quasi-Gray labelling.

Step 2c in Fig. 8 is the same as step 2c in Fig. 7.

Step 2d in Fig. 8, using this structure of G can guarantee small bit difference for at least (r +

2 )m neighbors for each CP. Any order of rows of G is feasible. The following sub-steps are to generate G as above in details

• Sub-step 2d-1) Set G₀ as an empty matrix, i = 0

• Sub-step 2d-2) Generate temporary matrix

• Sub-step 2d-3) If G_temp is full rank, then G_i+1 = G_temp, otherwise G_i+1 = G_;

• Sub-step 2d-4) if the rank of G_i+1 is not (r + 2 )m, go to sub-step 2c-5), otherwise, go to sub-step 2c-6).

• Sub-step 2d-5) i = i + 1, and go to sub-step 2c-2).

• Sub-step 2d-6) output G = G_i+1.

Steps 2a to 2d in Fig. 8 can in other words be formulated as that the linear mapping function / is based on a full rank matrix G, and wherein the rows or columns of the full rank matrix G includes complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an all-zero vector.

Step 2, 2a, 2b, 2c and 2d in Fig. 8 can be extended by using initial mapping from arbitrary bit vector B‘^nit to arbitrary generation vector B_g ^lnit. In this case, step 2, 2a, 2b, 2c and 2d are different from them in Fig. 8 and there is an additional step of 2e after 2d, which is not shown in Fig. 8.

Step 2, the first communication device 100 computes the intermediate bits as a bit vector B_g by a mapping from B_c , which can be written as B_g = (B_CG + B₀)mod2 or B_g = (B_CG - B₀)mod2.

Step 2a, B_g is generated based on B_g which is the generation vector. The mapping between B _g and B _g satisfies: for two arbitrary CPs (c_έ and x_;) with generation vector

their intermediate bit vector are

respectively. Given one neighbor of c_έ corresponding to intermediate bit vector

there must be a neighbor of x_; corresponding to the intermediate bit vector B_g ^~neis^{hbor )} satisfying:

Step 2b, the bit vector B‘^nit is mapped to generation vector B™^ic.

Step 2c, sort CPs based on the chordal distance to CP with generation vector B™^ic in ascending order. The corresponding generation vectors are recorded as

Step 2d, G is a binary matrix of size (r + 2 )m x (r + 2 )m, which can be expressed as:

and where the subset of CP generation vector , k_t >

0, is selected such that it minimizes the sum chordal distance from the CP with label

Step 2e (not shown in Fig. 8), B₀ = (B‘^nitG + B"^lit)mod2 is a constant.

Step 3 in Fig. 8, the B_g can be determined by B_g based on the table about the mapping between them. Another possible method is to determine B_g based on the B_g according to a predefined formulation. The reason why we need this step: if the property in step 2a is satisfied, we can generate G based on (r + 2)m neighbors. In this case, we can guarantee one bit difference between labels of any one CP and its first (r + 2 )m closest CPs. However, for the generation vector, only the last (r + 1 )m bits satisfy these properties, and therefore G is generated based on (r + 1 )m neighbors. In this case, we can only guarantee one bit difference between labels of any one CP and its first (r + 1 )m closest CPs. So, based on non-linear mapping in this step, for any one CP, bit difference from additional m closest CPs are optimized. Hence, method 2 performs equal or better to method 1.

Step 4 in Fig. 8 is the same as step 3 in Fig. 7.

Example of Method 2

In this example, the vector length of CP is 2² = 4, which means m = 2. The constellation size is 2⁵ = 32, r = 1/2. For B _g with last (r + 1 )m = 3 bits being

last 3 bits being [0,1,0], b_B [1,0] For B_g with last 3 bits being [0,0,1], b_Bfl = [1,1] For other cases, b_Bfl = [0,0] It can be proved that the property in step 2a is satisfied.

The bit vector [0,0, 0,0,0] is mapped to B_g = [0,0, 0,0,0] . Since r = 1/2 , (r + 2 )m = 5 intermediate bit vectors of neighbors from B_g = [0,0, 0,0,0] are needed. The neighbors of CP with B _g = [0,0, 0,0,0] are listed as follows in Table 5.

Table 5

In order to generate G, 5 of the CPs are needed. In this example, we select the 1^st ~5^th CPs. It should be noticed that other selections are also feasible. According to step 2d, the matrix G can be obtained as

-0 0 1 0 1-

1 0 1 0 1

G = 0 0 0 1 1

0 1 0 1 1

-0 0 1 1 1-

Then, for a given bit vector B_c, the corresponding generation vector can be determined by B_g = (B_cG)mod2. The results are listed as following in Table 6.

Table 6

If comparing the proposed indexing/labelling with generation vector based on indexing/ labelling and random indexing/ labelling, the average bit difference in average Hamming distance of indices/labels for constellation point pairs with minimum chordal distance is 1.667, 3, and slightly more than 3, respectively. It can be found that the average bit difference in average Hamming distance of labels for neighbors is the same as in method 1 , and it is significantly decreased compared to generation vector based on indexing/labelling and random indexing/labelling.

Indexing design for constellation generated based on second order Reed-Muller code with vector length of constellation point p^m . The solution shown in Fig. 7 and 8 can be extended to the case of vector length equal to p^m, where p is an odd prime number. In this case, the constellation point is generated based on the non-binary generation vector, i.e.

where l is a vector of length (r + 2 )m, and each element of l is in GF(p ) instead of GF( 2) as in Eq. (1). It should be noticed that if the desired constellation size is of the form N = 2^b , N constellation points are selected from the p^(r+2⁾m constellation points generated above. Then, r should satisfy r = ~

~ ^_ 2. Similar indexing method can be used for this case. The difference is that the input binary vector B_c should be mapped to a vector B_c with element in GF (p).

For example, method 1 can be implemented as follows and the corresponding flow chart is shown in Fig. 9.

Step 1 in Fig. 9, the first communication device 100 determines the B_c for modulation.

Step 2 in Fig. 9, the first communication device 100 determines the vector B_g for constellation generation by a mapping from B_c, which can be written as B_g = (B_cG)modp. The elements in B _g belong to GF(p ) instead of GF(2).ln other words the length N of the generated constellation symbol x is N = p^m, and wherein the linear mapping using the linear mapping function / is within a Galois field GF(p), where p is a prime number and m is a positive integer.

Step 2a in Fig. 9, vector B_c = [0,0, ... ,0] is mapped to generation vector

= [0,0, ... ,0]

Step 2b in Fig. 9, sort CPs based on the chordal distance to CP with generation vector

= [0,0, ... ,0] in ascending order. The corresponding generation vectors are recorded as

_R(l) 2) (pfr+2)m- 1)

Kg , K g_Q , .,. , K 9

Step 2c in Fig. 9, G is a binary matrix of size (r + 2 )m x (r + 2 )m, which can be expressed as:

where \_m is an m x m identity matrix, and g_t are the last (r + l)m elements in the vector B_g ^ki) where the subset of CP generation vectors , fc_£ > 0, is selected such that it

minimizes the sum chordal distance from the CP with index

Hence, steps 2a to 2c relates to the case when the linear mapping function / is based on a full rank block diagonal matrix G, where the rows or columns of at least one block of the full rank block diagonal matrix G includes part of complex number elements of vectors for generating constellation symbols. Mentioned constellation symbols are closest to the constellation symbol generated by an all-zero vector.

Step 3 in Fig. 9, the first communication device 100 determines the constellation point as a symbol vector x based on B_g according to Eq. (1) and (2).

With reference to the flow chart in Fig. 9 once more the following detailed steps are performed.

Step 1 in Fig. 9, the first communication device 100 obtains matrix B_c based on B_c according a pre-defined mapping between bits and values in GF(p).

Step 2 in Fig. 9 is the same as step 2 in Fig. 7 except for that mod2 operation is replaced by modp where p is the vector length of constellation point p^m as previously explained

Step 2a to step 2c in Fig.9 are the same as step 2a to step 2c in Fig.7, respectively.

It should be noticed that not all the p^(r+2)^m constellation points can be obtained according to the indexing above due to the limited range of input B_c. The method naturally selects part of the p^(r+2)m constellation points.

Since embodiments of the invention minimizes the number of different elements of B_c between neighbors, the mapping between B_c and B_c should minimize the bit difference for CP pair with element difference of B_c equals to 1. Therefore, when p > 2, and the first communication device 100 maps the bits for transmission B_c to a Galois field vector B_c, wherein the complex number elements of the Galois field vector B_c belong to {0,1, ... ,p - 1}; and maps the Galois field vector B_c using the linear mapping function / to obtain the vector B_g for generating a constellation symbol.

One example for mapping between B_c and B_c is as follows for the case of vector length equals to 3, N = 8, and r = 0 is given in Table 7.

Table 7

Indexing design for constellation generated based on second order Reed-Muller code with vector length of constellation point

- R '- ^{ln this} case, p_x < ··· < p_t, the second order

Reed-Muller code can still be used to generate the constellation point in the similar way as follow:

However, l in Eq. (3) is different from that in (1), where l can be expressed as follow

In Eq. (4), l is a vector with t parts, and each part u is composed by m_u elements. In the u- th part, each element

2, m._j}. In (3), the fc® is a vector with t parts, which is similar to l, i.e.,

In the ii-th part, each element is in GF(p_u ), i.e., k_{ e {0,1, ... ,p_u - 1], where / e ZJZ m_j +

In Eq. (3) P is a symmetric matrix of size åu₌₁m _t x å{,_{=1 ϊhέ}; and there are two types of P®, namely:

• P® type 1 : P® is block diagonal matrix, where the u- th block is a matrix of size m_u x m_u with elements belong to GF(p_u). The off-block diagonal entries are zeros.

• P® type 2: an extended structure of P®, where the off-block diagonal entries of P® in type 1 are multiples of the prime factor p_{l t} p₂, ..., or p_t. For example, the off-block diagonal entries of P® can be expressed as c - p_v , where c =

0,1, any value in {1,2, ... , t}.

Labelling method for P® type 1

This is an embodiment of method 3 explained previously. For P® type 1 , the follow indexing method can be utilized and the corresponding flow chart is shown in Fig. 10.

Step 1 in Fig. 10, the first communication device 100 determines the bits as a bit vector B_c for modulation.

Step 2 in Fig. 10, the first communication device 100 separates B_c into t parts, the u- th part includes [

bits.

Step 3 in Fig. 10, based on each part, the first communication device 100 determines the generation vectors B_{5 U} for a sub-constellation with symbol vector length

according to step 2 in Fig. 7 and step 2 in Fig. 9 which relates to the linear mapping using the linear mapping function /.

Step 4 in Fig. 10, the first communication device 100 determines the generation vector of the constellation B_g by combining all the sub-constellation generation vectors B_{5 U}.

Step 5 in Fig. 10, the first communication device 100 determines the symbol vector x based on B _g according to Eq. (3).

With reference to the flow chart in Fig. 10 once more the following detailed steps can be performed.

Step 1 in Fig. 10 is the same as step 1 in Fig. 7.

Step 2 in Fig. 10, any separation method and any order of these parts are feasible. Different separation methods or orders result in different orders of CPs in the constellation but keep the same set of CPs in the constellation. It should be noticed that the total number bits in B_c is at most å [log₂ , ^r“^t2)m"].

Steps 3 to 5 in Fig. 10, there are another two options, i.e. Option 1 (not shown in Fig. 10) comprising steps 3 to 6 and Option 2 (not shown in Fig. 10) comprising steps 3 and 4 to obtain symbol vector x. Step 3, based on each part, determine the generation vector B_{5 U} for a sub-constellation with symbol vector length p™^u according to methods in Fig. 7 or 9.

Step 4, for each sub-constellation, determine the P® and fc® according to the method in Fig. 6.

Step 5, determine the P® and fc® by: a. The ii-th block of P® is aP® , where a =

if p_u ¹ 2 , otherwise a =

b. The ii-th part of fc® js fc® .

Step 6, determine the symbol vector x based on P® and fc® according to Eq. (3).

Option 2

Step 3, based on each part, determine the symbol vector x_u from a sub-constellation with symbol vector length p™^u according to the method in Fig. 7 or 9.

Step 4, determine the symbol vector x as the Kronecker product of all x_u.

Example for P® Type 1

In this example, the vector length of CP is 2² x 3 = 12 , which means p = 2 ,p₂ = 3 ,m₁ = 2 ,m₂ = 1. In NR and LTE communication system, there are 12 subcarriers in one resource block (RB). Therefore, a symbol vector with length 12 can be mapped on one resource block in frequency domain and one OFDM symbol in time domain. The constellation size is 2⁸ = 256.

For any input bit vector B_c, the input bit vector can be divided into two parts, i.e.: B^ and B There are 5 bits in B^ and 3 bits in

B^ corresponds to a constellation (i.e. constellation 1) with 2⁵ CPs, where p = 2, and B ²⁾ corresponds to a constellation (i.e. constellation 2) with 3² CPs, where p₂ = 3.

According to the method 2 illustrated in Fig. 8, the generation vector of constellation 1 can be obtained, which is listed in the table in the example for method 2.

According to the method illustrated in Fig. 9, the generation vector of constellation 2 can be obtained as follow in Table 8.

Table 8

Therefore, the first 5 bits of generation vector are determined as in table 6 in the example for method 2, and the last 2 ternary value of generation vector are determined as in the table above. For example, if B_c = [1,0, 0,1, 1,1, 0,1], the generation vector is B_g = [0,1, 0,0, 1,2,0]

Then, the constellation point, i.e. symbol vector x, can be obtained based on the generation vector according to Eq. (3).

If comparing the proposed indexing with random indexing, the average bit difference in average Hamming distance of indices for constellation point pairs with minimum chordal distance is 1 .667 and around 4.5, respectively. It can be found that the average bit difference in average Hamming distance of indices for neighbors is significantly decreased compared to random index.

Labelling method for P⁽ type 2

This is an embodiment of method 3 explained previously.

For P® type 2, the off-block diagonal entries of P® can be non-zero values. Thus, comparing with P® type 1 , the maximum possible codebook size is larger. Similar indexing method as in Fig. 9 can be used.

The corresponding flow chart is shown in Fig. 11 .

Step 1 in Fig. 11 , the first communication device 100 separates the bits for transmission B_c into t parts, where the u- th part is B". Step 2 in Fig. 11 , the first communication device 100 maps each part of B" to a vector B" with element in GF p_u ), including the elements corresponding to off-blockdiagonal entries of P®

Step 3 in Fig. 11 , based on each vector B", the first communication device 100 determines the intermediate vector B_g from B_CG by mod(p_u ) for the u- th part of B_CG by linear mapping.

Step 2 and 3 can in other words be expressed as that the length N of the generated constellation symbol x is N =

- p™^c where p_t are prime numbers and m _t are positive integers for i = 1,2 ... t, where t is a positive integer, and wherein the linear mapping using the linear mapping function / for an i-th bit vector of the t number of vectors is within a Galois field GF(pi), where p_t is a prime number and m _t is a positive integer. Further, that when p_t > 2, the first communication device 100 is configured to map the i-th bit vector of the t number of vectors to a Galois field vector B®, wherein the complex number elements of the Galois field vector B® belong to {0,1, ... ,r_έ - 1}; and to map the Galois field vectors

using the linear mapping function / to obtain the vector B_g for generating a constellation symbol.

Step 3a in Fig. 11 , B_g is the generation vector. The one-to-one mapping between B_g and B_g satisfies the following property: for two arbitrary CPs c generation vector

and B^\ respectively, their intermediate bit vector are , respectively. Given one neighbor of c_έ corresponds to intermediate bit vector

there must be a neighbor of x_j corresponding to the intermediate bit vector g -^net.9^w,or) satisfying:

Step 3b in Fig. 11 , the bit vector B_c = [0,0, ... ,0] is mapped to

= [0,0, ... ,0] .

Step 3c in Fig. 11 , sort CPs based on the chordal distance to CP

Step 3d in Fig. 11 , G is a binary matrix of size (r + 2 )m x (r + 2 )m, which can be expressed

and where the subset of CP generation vectors k_t >

while at the same time guarantees that G is full rank, i.e., the rank of G is (r + 2 )m. This step is the same as step 2d in Fig. 8.

Step 4 in Fig. 11 , the first communication device 100 determines the generation vector of the constellation B_g by B_g = B_g®b_gg , where B_g is the intermediate vector and b_gg is the transformation matrix.

Step 5 in Fig. 11 , the first communication device 100 determines the symbol vector x based on B _g according to Eq. (3).

An example for symbol vector length equal to 12 is given as follow.

Example for P® Type 2

In this example, the vector length of CP is 2² x 3 = 12, which means p = 2 ,p₂ = 3 ,m₁ = 2 ,m₂ = 1. The symmetric matrix P® can be expressed as:

where P^ is the symmetric matrix with size 2 x 2 for a constellation with vector length 4 and size 2⁵. P^ is a symmetric matrix with size 1 x 1 (it is only a number) for a constellation with vector length 3 and size 3². P_off = [2 c_±, 2c₂]^T is a vector of off-block diagonal entries with size 2 x 1, where c ,c₂ e {0,1,2}. Comparing with the example for P® type 1 , | >g₂(3 x 3)J = 3 extra bits can be transmitted due to degree of freedom of c and c₂. The constellation size is

2¹¹ = 2048.

For a given input bit vector B_c, it is first mapped to a vector B_c with the structure:

where bc_t is binary and t_ci is ternary. The first 5 bits are the same for both B_c and B_c. The last 4 ternary numbers are determined as in Fig. 9. The intermediate vector B_g may be generated by:

B _g = [[B_cG]_{1 5}mod2, [B_cG]_{6 9}mod3] where G is a matrix of size 9 x 9, [BcG]^ is the vector of the first 5 elements of B_CG, and [B_CG]_6„9 is the 6^th~9^th elements of B_CG. Moreover, B_c = [0,0, 0,0, 0,0, 0,0,0] is mapped to B^ = [0,0, 0,0, 0,0, 0,0,0].

The generation matrix G may be obtained by the following steps:

1 . Bg is the generation vector which can be expressed as

b. where b_gi is binary and t_gi is ternary. [b_gl, b_g2, t_gl\ = fc® in (3), [b_g3, b_g4, b_g5] determines

c. Similar to the embodiment in Fig. 7section 2.2.22 , the first 2 binary numbers of B _g is defined as B^²⁾ = B^²⁾®b_Bfl, where the B^²⁾ and B^²⁾ are the first 2 bits of B _g and B₅, respectively. The value of b_Bfl satisfies which is the same as the embodiment in Fig. 7: d. For B _g with the 3^rd ~5^th bits being [1,0,0], b_Bfl = [0,1] e. For B _g with last 3^rd ~5^th bits being [0,1,0], b_Bfl = [1,0] f. For B _g with last 3^rd ~5^th bits being [0,0,1], b_Bfl = [1,1] g. For other cases, b_Bfl = [0,0]

2. Sort CPs based on the chordal distance to CP with

= [0,0, ... ,0] in ascending order.

The corresponding intermediate vectors are recorded as B^¹⁾, B^²⁾, ... , B^^29_1).

3. G is a matrix of size 9 x 9, which is expressed as:

b. where {#;} are the first 9 generation vectors in which

guarantee G is full rank, i.e., the rank of G is 9.

In this example, G can be expressed as: 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0

G = 0 0 0 0 1 1 1 0 1 1 0 1 1 2 0 2 0 0 1 0 2 1 2 0 0 0 2 1 0 1 0 2 0 0 0 2 -1 1 2 0 2 0 2 0 0

According to the mapping between

and B₅, the generation vector

can be obtained based on Finally, the constellation point, i.e. symbol vector x, can be determined based on generation vector according to Eq. (3).

If comparing the proposed index with random index, the average bit difference in average Hamming distance of indices for constellation point pairs with minimum chordal distance is 2.5 and around 6, respectively. It can be found that the average bit difference in average Hamming distance of indices for neighbors is significantly decreased compared to random index.

Performance Results

Through simulations it has been shown that embodiments of the invention have improved performance compared to conventional solutions.

One effect of embodiments of the invention is that the average bit difference in average Hamming distance of indices for constellation point pairs with minimum chordal distance is reduced by using the Quasi-Gray indexing solution according to the invention. The comparison is given in Table 9.

Table 9

Another effect of embodiments of the invention is that for a given signal-to-noise ratio (SNR) the bit error rate (BER) is reduced by using the Quasi-Gray indexing. On the other hand, for a given BER target, the required signal-to-noise ratio is reduced. In this simulation, two constellation points are transmitted on different OFDM symbols. The input bits B_c for generating constellation points are the same for the two constellation points, which is the information bits. So, no channel coding is used, or only repetition code is used.

The link-level simulation results for BER is shown in Fig. 12 in which x-axis shows signal to noise ratio (SNR) and y-axis shows block error rate (BLER). In this simulation, the typical scenario tapped-delay-line C (TDL-C) in NR is used. The simulation parameters can be found in Table 10.

From the simulation results, it can be found that 0.3dB SNR gain can be achieved by Quasi- Gray indexing according to the invention compared to random indexing, where the BER target is assumed to be 10^-2 and 10^-3.

Yet another effect of embodiments of the invention is that for a given SNR BLER is reduced by using the Quasi-Gray indexing according to the invention. On the other hand, for a given BLER target, the required signal-to-noise ratio is reduced. In communication systems like LTE and NR, information is considered to be correctly received only is all the bits in one block is correct. So, BLER is better for evaluating the invention than BER. In this simulation, multiple constellation points are transmitted on different OFDM symbols. The bits B_c for each constellation point is coded bit, where Polar code is used to generate B_c. The link- level simulation results for BLER is shown in Fig. 13 in which x-axis shows signal to noise ratio (SNR) and y-axis shows block error rate (BLER). The simulation parameters can be found in Table 11.

From the simulation results, it can be found that 0.3dB SNR gain can be achieved by Quasi- Gray indexing according to the invention compared to random indexing, where the BLER target is assumed to be 10^-3. Therefore, the transmitter power consumption can be reduced by using the Quasi-Gray indexing according to the invention. Moreover, if the Quasi-Gray indexing according to the invention is used, the coverage, defined as the region that can meet the BLER target, can be extended without increasing transmission power.

Furthermore, any method according to embodiments of the invention may be implemented in a computer program, having code means, which when run by processing means causes the processing means to execute the steps of the method. The computer program is included in a computer readable medium of a computer program product. The computer readable medium may comprise essentially any memory, such as a ROM (Read-Only Memory), a PROM (Programmable Read-Only Memory), an EPROM (Erasable PROM), a Flash memory, an EEPROM (Electrically Erasable PROM), or a hard disk drive.

Moreover, it is realized by the skilled person that embodiments of the first communication device 100 and the second communication device 300 comprises the necessary communication capabilities in the form of e.g., functions, means, units, elements, etc., for performing the solution. Examples of other such means, units, elements and functions are: processors, memory, buffers, control logic, encoders, decoders, rate matchers, de-rate matchers, mapping units, multipliers, decision units, selecting units, switches, interleavers, de- interleavers, modulators, demodulators, inputs, outputs, antennas, amplifiers, receiver units, transmitter units, DSPs, MSDs, TCM encoder, TCM decoder, power supply units, power feeders, communication interfaces, communication protocols, etc. which are suitably arranged together for performing the solution.

Especially, the processor(s) of the first communication device 100 and the second communication device 300 may comprise, e.g., one or more instances of a Central Processing Unit (CPU), a processing unit, a processing circuit, a processor, an Application Specific Integrated Circuit (ASIC), a microprocessor, or other processing logic that may interpret and execute instructions. The expression “processor” may thus represent a processing circuitry comprising a plurality of processing circuits, such as, e.g., any, some or all of the ones mentioned above. The processing circuitry may further perform data processing functions for inputting, outputting, and processing of data comprising data buffering and device control functions, such as call processing control, user interface control, or the like.

Finally, it should be understood that the invention is not limited to the embodiments described above, but also relates to and incorporates all embodiments within the scope of the appended independent claims.

Claims

1. A first communication device (100) for a communication system (500), the first communication device (100) being configured to obtain bits for transmission (B_c); obtain a vector (B₅) for generating a constellation symbol using a linear mapping function (/) based on the bits for transmission (B_c); and generate a constellation symbol (x) based on the vector ( B₅ ) for generating a constellation symbol, wherein the generated constellation symbol (x) comprises multiple complex number elements.

2. The first communication device (100) according to claim 1 , wherein the linear mapping function (/) is based on a full rank block diagonal matrix (G) comprising at least two blocks, wherein one of the at least two blocks is an identity matrix, and wherein the rows or columns of at least one block of the full rank block diagonal matrix (G) includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an all-zero vector.

3. The first communication device (100) according to claim 1 , configured to map the bits for transmission (B_c) using the linear mapping function (/) to obtain an intermediate vector (B₅); obtain a transformation vector (b_Bfl) having m complex number elements based on the intermediate vector (B₅), where p is a prime number and p^m is the length of the constellation symbol (x); and obtain the vector (B₅) for generating a constellation symbol based on an XOR operation of the intermediate vector (B₅) and the transformation vector (b_Bfl).

4. The first communication device (100) according to claim 3, wherein the linear mapping function (/) is based on a full rank matrix (G), and wherein the rows or columns of the full rank matrix (G) includes complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an allzero vector.

5. The first communication device (100) according to any one of the preceding claims, wherein the length N of the generated constellation symbol (x) is N = p^m, and wherein the linear mapping using the linear mapping function (/) is within a Galois field GF(p), where p is a prime number and m is a positive integer.

6. The first communication device (100) according to claim 5, wherein p > 2 , and further configured to map the bits for transmission (B_c) to a Galois field vector (B_c), wherein the complex number elements of the Galois field vector (B_c ) belong to (0,1, ... , p - 1}; and map the Galois field vector (B_c) using the linear mapping function (/) to obtain the vector (B₅) for generating a constellation symbol.

7. The first communication device (100) according to any one of the preceding claims, configured to separate the bits for transmission (B_c) to obtain t number of vectors, wherein t is a positive integer dependent on the length N of the constellation symbol (x); map the t number of vectors using the linear mapping function (/) to obtain t number of symbols (x*) for generating a constellation symbol; and generate the constellation symbol (x) based on the t number of symbols (x*) for generating a constellation symbol.

8. The first communication device (100) according to claim 7, wherein the constellation symbol (x) is the Kroneck product of the t number of symbols (x*) for generating a constellation symbol.

9. The first communication device (100) according to claim 7 or 8, wherein the linear mapping function (/) is based on a full rank block diagonal matrix G, and wherein the rows or columns of at least one block of the full rank block diagonal matrix G includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to the constellation symbol generated by an all-zero vector.

10. The first communication device (100) according to any one of claims 7 to 9, wherein the length N of the generated constellation symbol (x) is N =

... r™⁽ where p_t are prime numbers and m _t are positive integers for i = 1,2 ... t, where t is a positive integer, and wherein the linear mapping using the linear mapping function (f) for an i-th bit vector of the t number of vectors is within a Galois field GF(p_j), where p_t is a prime number and m _t is a positive integer.

11. The first communication device (100) according to claim 10, wherein p_t > 2, and further configured to map the i-th bit vector of the t number of vectors to a Galois field vector (B^ ), wherein the complex number elements of the Galois field vector (B^) belong to (0,1, ... , r_έ - 1}; and map the Galois field vectors (B^) using the linear mapping function (/) to obtain the vector (B₅) for generating a constellation symbol.

12. A second communication device (300) for a communication system (500), the second communication device (300) being configured to receive a symbol (Y) mapped onto time-frequency resources; demodulate the received symbol (Y) based on a symbol constellation, wherein the symbol constellation comprises multiple constellation symbols, each constellation symbol (x) comprising multiple complex number elements and wherein each constellation symbol (x) is generated based on a vector (B₅) for generating a constellation symbol, wherein the vector (B₅) for generating a constellation symbol is obtained by using a linear mapping function (f) based on possible transmitted bits (B_c).

13. The second communication device (300) according to claim 12, wherein the linear mapping function (/) is based on a full rank block diagonal matrix (G) comprising at least two blocks, wherein one of the at least two blocks is an identity matrix, and wherein the rows or columns of at least one block of the full rank block diagonal matrix (G) includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an all-zero vector.

14. The second communication device (300) according to claim 12, wherein the vector (B₅) for generating a constellation symbol is based on an XOR operation of an intermediate vector (B₅) and a transformation vector (b_Bfl), wherein the transformation vector (b_Bfl) has m complex number elements based on the intermediate vector (B₅), where p is a prime number and p^m is the length of the constellation symbol (x), and wherein the intermediate vector (B₅) is obtained by mapping possible transmitted bits (B_c) using the linear mapping function (/).

15. The second communication device (300) according to claim 14, wherein the linear mapping function (/) is based on a full rank matrix (G), and wherein the rows or columns of the full rank matrix (G) includes complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to a constellation symbol generated by an allzero vector.

16. The second communication device (300) according to any one of claims 12 to 15, wherein the length N of the constellation symbol (x) is N = p^m, and wherein the linear mapping using the linear mapping function (/) is within a Galois field GF(p), where p is a prime number and m is a positive integer.

17. The second communication device (300) according to claim 16, wherein p > 2 , and wherein the possible transmitted bits (B_c) are mapped to a Galois field vector (B_c ), wherein the complex number elements of the Galois field vector (B_c ) belong to (0,1, ... ,p - 1], and wherein the vector (B₅) for generating a constellation symbol is obtained by mapping the Galois field vector (B_c ) using the linear mapping function (/).

18. The second communication device (300) according to any one of claims 12 to 17, wherein the constellation symbol (x) is based on t number of symbols (x*) for generating a constellation symbol, where t is a positive integer dependent on the length N of the constellation symbol (x), and wherein the t number of symbols (x*) for generating a constellation symbol is obtained by mapping t number of vectors using the linear mapping function (/), wherein the t number of vectors is a separation of the possible transmitted bits (B_c).

19. The second communication device (300) according to claim 18, wherein the constellation symbol (x) is the Kroneck product of the t number of symbols (x*) for generating a constellation symbol.

20. The second communication device (300) according to claim 18 or 19, wherein the linear mapping function (/) is based on a full rank block diagonal matrix G, and wherein the rows or columns of at least one block of the full rank block diagonal matrix G includes part of complex number elements of vectors for generating constellation symbols, wherein the constellation symbols are closest to the constellation symbol generated by an all-zero vector.

21. The second communication device (300) according to any one of claims 18 to 20, wherein the length N of the constellation symbol (x) is N = p™ ™² ... r™⁽ where p_t are prime numbers are positive integers for i = 1,2 ... t, where t is a positive integer, and wherein the linear mapping using the linear mapping function (/) for an i-th bit vector of the t number of vectors is within a Galois field GF(pi), where p_t is a prime number and

is a positive integer.

22. The second communication device (300) according to claim 21 , wherein p_t > 2 , and wherein the vector (B₅) for generating a constellation symbol is obtained by mapping Galois field vectors (B^) using the linear mapping function (/), wherein the Galois field vectors (B^) is obtained by mapping the i-th bit vector of the t number of vectors to the Galois field vector (B^), wherein the elements of the Galois field vector (B^) belong to {0,1, - ,Vi ~ 1}·

23. A method for a first communication device (100), the method (200) comprising obtaining (202) bits for transmission (B_c); obtaining (204) a vector (B₅) for generating a constellation symbol using a linear mapping function (/) based on the bits for transmission (B_c); and generating (206) a constellation symbol (x) based on the vector (B₅) for generating a constellation symbol, wherein the generated constellation symbol (x) comprises multiple complex number elements.

24. A method for a second communication device (300), the method (400) comprising receiving (402) a symbol (Y) mapped onto time-frequency resources; demodulating (404) the received symbol (Y) based on a symbol constellation, wherein the symbol constellation comprises multiple constellation symbols, each constellation symbol (x) comprising multiple complex number elements and wherein each constellation symbol (x) is generated based on a vector (B₅) for generating a constellation symbol, wherein the vector (B₅) for generating a constellation symbol is obtained by using a linear mapping function (f) based on possible transmitted bits (B_c).

25. A computer program with a program code for performing a method according to claim 23 or 24 when the computer program runs on a computer.