CA2238015A1 - A method and an arrangement for making fixed length codes - Google Patents
A method and an arrangement for making fixed length codes Download PDFInfo
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Abstract
The present invention relates to a method in a telecommunications system comprising transmission of information over a channel and encoding of the transmitted information wherein a fixed length code is designed according to the following steps: initialising of a code table, wherein a code word, hereafter called the kernel, is allocated to a position corresponding to a peak of a probability density function of the transmitted information and said kernel can be chosen arbitrarily as a n-bit code word; generating n new code words with the Hamming distance one to said kernel; packing said new code words as close as possible on either side of the kernel.
Description
CA 0223801~ 1998-0~
WO 97/2039S . PCT/SE96/01555 A ~Jl~HoD AND AN ARRANGEMENT FOR MAK~NG FIXED LENGTH COD~S
TECHNICAL FIELD
The present invention relates to a method of making fixed length codes more robust against bit errors. The invention also rel~tes to an arrangement for carrying out the method.
DESCRIPTION OF THE BACKGROUND ART
Events can be coded in binary code words. A binary encoding method with substantially uniform rate is shown in the US patent US 5,300,930. Often one wants to code the information in as few bits as possible in order to minimise the bandwidth required. Depending on the probability distribution of the events it is possible to code the information in such a way that events that occur rarely are coded with longer code words and events that occur frequently are coded with shorter code words and the encoding is done so that the decoder can distinguish between the different code words.
Such codes are called instantaneous.
Very efficient codes can be designed in this way and the method of coding is called entropy coding.
However, assume that the transmission channel between the encoder and decoder introduces bit errors. In this case probiems arise because the decoder cannot with certainty decode the different variable length code words since it is not possible to see where one code word starts and CA 0223801~ 1998-0~
WO 97/2039S . PCT/SE96/01555 A ~Jl~HoD AND AN ARRANGEMENT FOR MAK~NG FIXED LENGTH COD~S
TECHNICAL FIELD
The present invention relates to a method of making fixed length codes more robust against bit errors. The invention also rel~tes to an arrangement for carrying out the method.
DESCRIPTION OF THE BACKGROUND ART
Events can be coded in binary code words. A binary encoding method with substantially uniform rate is shown in the US patent US 5,300,930. Often one wants to code the information in as few bits as possible in order to minimise the bandwidth required. Depending on the probability distribution of the events it is possible to code the information in such a way that events that occur rarely are coded with longer code words and events that occur frequently are coded with shorter code words and the encoding is done so that the decoder can distinguish between the different code words.
Such codes are called instantaneous.
Very efficient codes can be designed in this way and the method of coding is called entropy coding.
However, assume that the transmission channel between the encoder and decoder introduces bit errors. In this case probiems arise because the decoder cannot with certainty decode the different variable length code words since it is not possible to see where one code word starts and CA 0223801~ 1998-0~
another ends. Of course, this can have a devastating effect on the interpretation of the received information.
Usually this problem is solved by means of channel coding, where controlled redundancy is introduced in the channel encoder in such a way that the channel decoder can detect and correct erroneous bits.
Using this method more bits have to be transmitted over the channel and the channel bit rate increases as well as the complexity due to the channel encoder and decoder. Moreover, at bit error rates that are higher than the error correcting capacity of the channel coding, the decoded bit error rate often becomes worse than the uncoded bit error rate. Such a threshold exists for every code, either limited by the Shannon capacity formula or on a restriction in acceptable delay due to channel coding.
One way to overcome some of the drawbacks with a variable length code, in terms of bit error robustness, is to use a fixed length code. Here we lose some efficiency because all code words have the same length independent of the probability distribution of the events to be coded. However, once synchron;sed, each code word can be decoded independent of any other code word and this makes the code rather robust.
If we assume that the information to be transmitted represents a 'level' monotonically increasing from a minimum value to a maximum value the events represent discrete points on this level curve and each point is coded with a fixed length code word.
A bit error in a code word causes a jump in the level curve and dependent on which bit in the code word that is erroneous, the level jump becomes smaller or larger. This means that some bit errors (at certain positions in the code word) may cause an acceptable decoded level (small level error) while other error bits in a code word cause a totally unacceptable level change.
CA 0223801~ 1 998 - o~
wo 97/20395 3 PCT/SE96/01555 The common way to improve the robustness to bit errors of a fixed length code table is by means of Gray coding is to be found in the following two books:
[1] ~ohn A.C. Bingham, "The Theory and Practice of Modem Design", John Wiley & Sons, 1988 (pp. 65-66).
[2] E.A.Lee, D.G. Messerschmitt, nDigital Communication", Kluwer Academic Publishers Boston, 1988 (pp. 193).
A well known method to decrease the bit error sensitivity of a fixed length code is by means of Gray coding which is shown in figure 1. Gray codes ' have the property that nearest neighbours are encoded into code words that differ by only one bit, i.e. the Hamming distance is 1. This works fine for a 2-bit code 14 words) in a 2-dimensional signal constellation, where also the Euclidean distance is at a minimum for code words with Hamming distance = 1 .
In a 1-dimensional representation, when the code words represent a level, the situation is different, as illustrated with the following 3-bit codes with 8levels .
A straight forward code and two Gray codes with associated bit error distances are shown in Table 1. The distance d1 is the sum of the distances obtained by changing one bit at a time in the code word. The mean distance per bit is then d1 / 3 units.
WO g7/20395 4 PCT/SE96101555 Level Straight Code Gray Code 1 Gray Code 2 Code Disl~nce dl Code Distance d1 Code Dislal~ce dl 0 000 4+2+1=7 000 ~+1+3=9 0003+1+7=11 1 001 4~2+1=7 010 5+1+1=7 0101+1+5=7 2 010 4+2+1=7 011 5+1+1=7 1101+1+3=5 3 011 4+2+1=7 001 1+1+3=5 1003+1+1=5 4 100 4+2+1=7 101 1+3+1=5 1013~1~1=5 6 101 4+2+1=7 100 5+1+1=7 1111+1+3=5 6 110 4+2+1=7 110 5+1+1=7r0111+1+5=7 7 111 4+2+1=7 111 5+3+~=9'0013+1+7=11 Table 1.
Using a metric based upon the Euclidean distance, d1, or the squared Euclidean distance, d2, we can define the following distances 1 M-l p(m) dl(m) n m~O
and 1 M-l ~2 =--~ p(m) d2(m) n m=O
1 0 where n is the number of bits in code word, M is the number of code words, plm) is the pdf (probability density function) of the level, m = 0, 1, ..., M-1,el is the mean error distance per bit and ~2 iS the mean squared distance (mse) per bit.
The mean error distance per bit is ~l = 713=2~33 units for all three codes.
Thus, for a uniform pdf for the level, nothing is gained with a Gray code compared with the straight code. However, if the levels 3 and 4 (2,3,4 and ~;) are more common than the other levels a gain of the Gray code 1 (Gray code 2) in the mean error distance is achieved.
CA 0223801~ 1998-0~
Usually this problem is solved by means of channel coding, where controlled redundancy is introduced in the channel encoder in such a way that the channel decoder can detect and correct erroneous bits.
Using this method more bits have to be transmitted over the channel and the channel bit rate increases as well as the complexity due to the channel encoder and decoder. Moreover, at bit error rates that are higher than the error correcting capacity of the channel coding, the decoded bit error rate often becomes worse than the uncoded bit error rate. Such a threshold exists for every code, either limited by the Shannon capacity formula or on a restriction in acceptable delay due to channel coding.
One way to overcome some of the drawbacks with a variable length code, in terms of bit error robustness, is to use a fixed length code. Here we lose some efficiency because all code words have the same length independent of the probability distribution of the events to be coded. However, once synchron;sed, each code word can be decoded independent of any other code word and this makes the code rather robust.
If we assume that the information to be transmitted represents a 'level' monotonically increasing from a minimum value to a maximum value the events represent discrete points on this level curve and each point is coded with a fixed length code word.
A bit error in a code word causes a jump in the level curve and dependent on which bit in the code word that is erroneous, the level jump becomes smaller or larger. This means that some bit errors (at certain positions in the code word) may cause an acceptable decoded level (small level error) while other error bits in a code word cause a totally unacceptable level change.
CA 0223801~ 1 998 - o~
wo 97/20395 3 PCT/SE96/01555 The common way to improve the robustness to bit errors of a fixed length code table is by means of Gray coding is to be found in the following two books:
[1] ~ohn A.C. Bingham, "The Theory and Practice of Modem Design", John Wiley & Sons, 1988 (pp. 65-66).
[2] E.A.Lee, D.G. Messerschmitt, nDigital Communication", Kluwer Academic Publishers Boston, 1988 (pp. 193).
A well known method to decrease the bit error sensitivity of a fixed length code is by means of Gray coding which is shown in figure 1. Gray codes ' have the property that nearest neighbours are encoded into code words that differ by only one bit, i.e. the Hamming distance is 1. This works fine for a 2-bit code 14 words) in a 2-dimensional signal constellation, where also the Euclidean distance is at a minimum for code words with Hamming distance = 1 .
In a 1-dimensional representation, when the code words represent a level, the situation is different, as illustrated with the following 3-bit codes with 8levels .
A straight forward code and two Gray codes with associated bit error distances are shown in Table 1. The distance d1 is the sum of the distances obtained by changing one bit at a time in the code word. The mean distance per bit is then d1 / 3 units.
WO g7/20395 4 PCT/SE96101555 Level Straight Code Gray Code 1 Gray Code 2 Code Disl~nce dl Code Distance d1 Code Dislal~ce dl 0 000 4+2+1=7 000 ~+1+3=9 0003+1+7=11 1 001 4~2+1=7 010 5+1+1=7 0101+1+5=7 2 010 4+2+1=7 011 5+1+1=7 1101+1+3=5 3 011 4+2+1=7 001 1+1+3=5 1003+1+1=5 4 100 4+2+1=7 101 1+3+1=5 1013~1~1=5 6 101 4+2+1=7 100 5+1+1=7 1111+1+3=5 6 110 4+2+1=7 110 5+1+1=7r0111+1+5=7 7 111 4+2+1=7 111 5+3+~=9'0013+1+7=11 Table 1.
Using a metric based upon the Euclidean distance, d1, or the squared Euclidean distance, d2, we can define the following distances 1 M-l p(m) dl(m) n m~O
and 1 M-l ~2 =--~ p(m) d2(m) n m=O
1 0 where n is the number of bits in code word, M is the number of code words, plm) is the pdf (probability density function) of the level, m = 0, 1, ..., M-1,el is the mean error distance per bit and ~2 iS the mean squared distance (mse) per bit.
The mean error distance per bit is ~l = 713=2~33 units for all three codes.
Thus, for a uniform pdf for the level, nothing is gained with a Gray code compared with the straight code. However, if the levels 3 and 4 (2,3,4 and ~;) are more common than the other levels a gain of the Gray code 1 (Gray code 2) in the mean error distance is achieved.
CA 0223801~ 1998-0~
SUMMARY OF THE INVENTION
This invention is about how to make fixed length codes more robust against bit errors, in cases when the probability density function (pdf) has a peak ~not uniformly distributed). A considerable gain in mean squared error ~mse) can be achieved in these cases, and the actual gain depends on the number of bits in the code word and the actual pdf of the level.
PROBLEM DESCRIPTION
Introducing a bit error in the straight code in Table 1, the level jump may be 1, 2 or 4 units depending on the position of the erroneous bit. This is valid for each of the 8 code words and the mean error distance is ~l = 7/3 =
2.33 units per bit. If the mse is used, the figure becomes ~2 = ~12+22+42)/3 = 7 units2 per bit. These values on ~1 and ~2 are independent of the actual pdf, and therefore this straight code may serve as a reference for other codes.
Assuming a uniform pdf for the Gray code 1 we get a mean error distance of ~1 = 2*~9+7+7+5)/(8*3) = 7/3 = 2.33 units per bit which is equal to the straight coding. The mse becomes ~2 = 2~i/3 = 8.33 units2 per bit.
The corresponding figures for the Gray code 2 are; mean error distance ~1 =
7/3 = 2.33 units per bit and mse ~2 = 27/3 = 9 units2 per bit.
Thus, nothing is gained by using Gray coding compared to straight coding if the level pdf is uniform. On the contrary, if mse is chosen as the metric there is a performance loss.
However, if we have a pdf with a peak in the central part of the coding table a performance gain can be anticipated with the Gray code tables.
CA 0223801~ 1998-0~
This invention is about how to make fixed length codes more robust against bit errors, in cases when the probability density function (pdf) has a peak ~not uniformly distributed). A considerable gain in mean squared error ~mse) can be achieved in these cases, and the actual gain depends on the number of bits in the code word and the actual pdf of the level.
PROBLEM DESCRIPTION
Introducing a bit error in the straight code in Table 1, the level jump may be 1, 2 or 4 units depending on the position of the erroneous bit. This is valid for each of the 8 code words and the mean error distance is ~l = 7/3 =
2.33 units per bit. If the mse is used, the figure becomes ~2 = ~12+22+42)/3 = 7 units2 per bit. These values on ~1 and ~2 are independent of the actual pdf, and therefore this straight code may serve as a reference for other codes.
Assuming a uniform pdf for the Gray code 1 we get a mean error distance of ~1 = 2*~9+7+7+5)/(8*3) = 7/3 = 2.33 units per bit which is equal to the straight coding. The mse becomes ~2 = 2~i/3 = 8.33 units2 per bit.
The corresponding figures for the Gray code 2 are; mean error distance ~1 =
7/3 = 2.33 units per bit and mse ~2 = 27/3 = 9 units2 per bit.
Thus, nothing is gained by using Gray coding compared to straight coding if the level pdf is uniform. On the contrary, if mse is chosen as the metric there is a performance loss.
However, if we have a pdf with a peak in the central part of the coding table a performance gain can be anticipated with the Gray code tables.
CA 0223801~ 1998-0~
Still, the minimum distance and the minimum squared distance are for the Gray code tables 5/3 = 1.67 units per bit and 1113 = 3.67 units2 per bit, respectively, and these values may be too large to be acceptable and they do not represent the minimum values that can be achieved.
Sometimes the pdf does not have a peak at the central part but may have another shape, for example, two peaks on either side of the central part of the level curve. To cope with this situation another design method is required to build a bit error robust fixed length code tsble.
The chrominance ~colour difference~ components represented by the fixed length tables usually have a very pronounced peak, therefore the invented code should be suitable for improving the robustness to bit errors in this context .
BEST MODES OF CARRYING OUT THE INVENTION
A mobile telephone radio communication system is shown in figure 2. A radio base station BS transmits information to and from a mobile radio terminal MS. The radio connection is shown in the figure as a zig-zag symbol between the radio base station BS and the mobile radio terminal MS.
Figure 3 is a flow chart illustrating how information is transferred from the radio base station BS to the mobile radio terminal MS of figure 2. A block 101 illustrates an input device which in the embodiment is e.g. a video camera connected to the radio base station shown in figure 2. A block 10Z
illustrates a digitiser connected to the input device. An encoder 103 is in accordance with the invention connected to the digitiser. A transmitter is illustrated with block 104. The blocks 102-104 are parts of the radio base station BS. Block 105 illustrates the channel on which information is transmitted from the radio base station BS to the mobile radio terminal MS.
The channel is a radio channel of 9.6 kbits/s disturbed by e.g. Rayleigh , CA 0223801~ 1998-0~
Sometimes the pdf does not have a peak at the central part but may have another shape, for example, two peaks on either side of the central part of the level curve. To cope with this situation another design method is required to build a bit error robust fixed length code tsble.
The chrominance ~colour difference~ components represented by the fixed length tables usually have a very pronounced peak, therefore the invented code should be suitable for improving the robustness to bit errors in this context .
BEST MODES OF CARRYING OUT THE INVENTION
A mobile telephone radio communication system is shown in figure 2. A radio base station BS transmits information to and from a mobile radio terminal MS. The radio connection is shown in the figure as a zig-zag symbol between the radio base station BS and the mobile radio terminal MS.
Figure 3 is a flow chart illustrating how information is transferred from the radio base station BS to the mobile radio terminal MS of figure 2. A block 101 illustrates an input device which in the embodiment is e.g. a video camera connected to the radio base station shown in figure 2. A block 10Z
illustrates a digitiser connected to the input device. An encoder 103 is in accordance with the invention connected to the digitiser. A transmitter is illustrated with block 104. The blocks 102-104 are parts of the radio base station BS. Block 105 illustrates the channel on which information is transmitted from the radio base station BS to the mobile radio terminal MS.
The channel is a radio channel of 9.6 kbits/s disturbed by e.g. Rayleigh , CA 0223801~ 1998-0~
7 PCT/SE:96/01555 fading. Block 106 illustrates a receiver situated in the mobiie radio terminal MS and a decoder is illustrated by block 107. Block 108 symboiises e.g. a video monitor. The input device 101 comprises an information source that generates a variable that is a function of time F(t), wherein a probability density function for values of F(t) occurring in said information source has a maximum. The digitiser 102 and the encoder 103 of the function F(t) perform digitisation and encoding of F(t) into a function F'(t) by using a tablesuch that the Hamming distance between code words of F'(t) that are adjacent to one another is lower for values of F(t) that are near the maximum of the probability density function than for values of Flt) that are further away from said maximum. The transmitter 104 transmits the information in the function F'(t) over the channel 105. The information is received by the receiver 106 as F"(t) because F'~t) has been disturbed by Rayleigh fading in the channel 105. The decoder 107 decodes the received function F"(t) back into a function F'"(t) which is similar to the original function F(t), by using a table which is the inverse of the table in the encoder 103.
The earlier mentioned Gray code is characterised by the fact that the Hamming distance for a code word to the closest neighbours is one. If the code represents a level monotonicaliy increasing from a minimum value to a maximum value the Hamming distance may not be the proper metric to use for a robust code design. Rather, one should use a metric that reflects the level change when the different bits in a code word change, such as the Euclidean distance or the squared Euclidean distance (the mse).
The design method for the pdf with a peak is illustrated with a 3 bit code table, the so called Baang-1 code. We assume that we know the pdf or at least the position of the peak of the pdf representing the level curve.
The other code design method, suitable for more general pdf:s is illustrated by a 4 bit code table, the so called Baang-2 code.
CA 0223801~ 1998-0~
The earlier mentioned Gray code is characterised by the fact that the Hamming distance for a code word to the closest neighbours is one. If the code represents a level monotonicaliy increasing from a minimum value to a maximum value the Hamming distance may not be the proper metric to use for a robust code design. Rather, one should use a metric that reflects the level change when the different bits in a code word change, such as the Euclidean distance or the squared Euclidean distance (the mse).
The design method for the pdf with a peak is illustrated with a 3 bit code table, the so called Baang-1 code. We assume that we know the pdf or at least the position of the peak of the pdf representing the level curve.
The other code design method, suitable for more general pdf:s is illustrated by a 4 bit code table, the so called Baang-2 code.
CA 0223801~ 1998-0~
8 PCTtSE96/01555 7.1 Pdf with a peak (Baang-1 code) Here we want to minimise the distances for code words taken from a particular code word, hereafter denoted the kernel, and outwards. The code S words are taken in an order of increasing distance from the kernel.
7.1 .1 ) Initialise the code table.
7.1.1 .a) Pick the kernel, for example 111, and put it on that level position that corresponds to the peak of the level pdf, in our case, say level 3.
Level Code Word 3 111 (kernel) 7.1.1.b) On either side of the kernel, insert two new code words with the Hamming distance one to the kernel. In this case we change the leftmost bit first and then the next bit in order. However, any order of bitchange can be used .
Level Code Word Distance d1 3 111 1 +1 +?=?
7.1.1.c) The third bit, the rightmost bit of the kernel, is now changed and the new code word is put on level 1 (or level 5) Level Code Word Distance d1 d2 3 111 1+1+2=412+12+22=6 CA 0223801~ 1998-0~
With this initialisation we have packed all code words with the Hamming distance one as close as possible ~in the ~uclidean distance sense) to the kernel .
Should there be more than 3 bits in the code words, say n bits, the same procedure is applied and the result is an initialisation of (n + 1 ) code words.
7.1.2) Continue to build the table.
7.1 .2.a) The next code word to consider is the one closest to the kernel and with the maximum degree of freedom to design new code words in a compact way.
Level 2 with the code word 011 has code words on each side, but level 4 with the code word 101 has a code word only on one side (111), and thus has the maximum degree of freedom. This code word (101) is called the 1 :st reference.
Thus, changing the bits in the 1 :st reference we get the following table:
(here we use the same order for changing bits as was used in the initialisation, but any other order can be used as well) Level Code Word Distance d1 d2 3 111 1 +1+2--4 6 4 --> 101 1 t1 tZ=4 6 CA 02238015 1998-0~-15 W O ~712Q395 10 PCT~SE96/01555 If not the bit changed code word is in the table we add it as close as possible to the reference word.
7.1.2.b) Following the principle in 2a) we now consider level 2 and choose code word 011 as the 2:nd reference and get Level Code Word Distance dl d2 2 --> 011 1+3+2--6 14 3 111 1+1 +2=4 6 4 101 1 + 1+2=4 6 7.1 .2.c) For the next reference to consider (110 or 001) there is no difference in the distance to the kernel and the number of freedom is the same and so 2a) does not give us the information how to proceed and any side can be used.
In this example we stick to the symmetry principle and process that code word that is on the opposite side of the kernel of the last processed code word.
In this case it means level 6 with the reference word 001, and the result is Level Code Word Distance d1 d2 2 011 1 +3+2z614 3 111 1+1+234 6 4 101 1+ 1+2--4 6 --> 001 1+3+2=6 14 CA 0223801~ 1998-0~
7.1.3) All 8 code words are now associated with a level and we just calculate the distance d1 (m) and mse d2(m) for the remaining 4 code words.
~, Table 2 shows the final result, the 3 bit Baang-1 code.
Straight Code Baang-1 Code Code Distance Code Distance dl d1 d2 o 000 4+2+1 =7 010 1 +7+2=10 54 001 4+2+1=7 110 1+5+2=8 30 2 010 4+2+1=7 011 1+3+2=6 14 3 011 4+2+1=7 111 1+1+2=4 6 4 100 4+2+1 =7 101 1 +1 +2-4 6 101 4+2+1 =7 001 1 +3+2=6 14 6 110 4+2+1=7 100 1+5+2=8 30 7 111 4+2+1=7 000 1+7+2=10 54 Table 2.
As can be seen from the table, the distance d2 for the straight code is 21 for each code word.
7.1.3) Te~ .alion of the table.
If a new code word designed from a reference and allocated according to the principles outlined above falls outside one of the end levels, a reference word from the other side of the kernel is used. This may happen, for example, when the peak of the pdf is not in the central part of the level distribution. The following example illustrates the principle, where we assume that the peak of the pdf is located at level 5.
CA 022380l5 l998-05-l5 Step 1 Step 2 Step 3 Step 4 Level Code Code Code CodeDistance d1 0 0001 +1+4=6 1 100 1001 +6+5=12 2 010 010 0105+2+2=9 3 001 001 -->0013+1+3=7 4 --> 011 011 0111 ~1+2=4 111 111 111 1111 +1+2=4 6 101 --> 101 1013~1+5=9 7 110 110 110~;+6+2=13 7.2 More general pdf (Basng-2 code).
This technique may be used when the pdf has a more general shape and the design steps are as follows. Here we illustrate the method by a 4 bit code (n = 4) .
7.2.1) Initialise the code tsble.
7.2.1.a) Pick a (n-1) bit code word, for example 111, denoted kernel, and put it on the level position that corresponds to the peak of the level pdf, say level 8.
Level Code Word 7.2.1.b) Insert a maximum of ~n-1) new code words with Hamming distance one from the kernel and put them on the distances d = ld1, d2, ... dk ~ units from the kernel, in our example we choose dt = 1, d2 = 3 and d3 = 5 and we get, -CA 022380l5 l998-05-l5 WO 97/20395 PCT/SE96/015~5 Level Code Word __ 5 101 d3 6 d2 ~~~~i~ 7 110 -- 8 111 (kernel~
~0 7.2.Z) Continue to build the table.
7.2.2.a) The code table is built from the kernel and upwards and the code word associated with the distance d1 is used as the 1 :st reference, i.e. 110.
(The index in the distance measure is taken as the order in which the references are processed).
Changing, the bits in the 1 :st reference and packing the new words as close as possible to the kernel we get Level Code Word 7 --> 110 Next we use the code word associated with d2, 101, as a reference, 3 0 changing the bits and obtain W O 97/20395 14 PCT/S~96101555 Level Code Word 5 -> 101 Next reference corresponding to d3, 01 1, does not give any new code word, and all the references have now been used.
~5 2.2.2~b) With the code word next to the kernel as a reference, 1 10, chan~e the bits and fili up the table with new code words in positions as close as possible to the kernel. In this case 1 10 does not give any new code word and so continue with the next code word counted from the kernel, namely 010.
Level Code Word 6 --~ 010 and we have all 8 code words.
CA 0223801~ 1998-0~
7.2.2.c) To obtain the 16 possible code words for the 4 bit code we take the mirror picture of the 8 code words below the 8:th word (or above the 1 :st word) and put a 1 (or 0) at the leftmost position (or rightmost position or any position) of the 8 original words and a 0 (or 1 ) at the same bit position in the reflected 8 words,.i.e. one implementation gives Level Code WordDistance d1 The design method ensures a symmetrical distance distribution d1 ~m) and d2(m) and it allows a flexibility to adapt the distance distribution to the actual pdf of the level in such a way that a small distance is allocated to the peak(s) of the pdf, thereby making the mean error distance per bit ~l and the squared distance per bit ~2 small.
, 30 Forbidden code words are removed from the table.
CA 022380l5 l998-05-l5 To illustrate the flexibility of the Baang-2 code methodolo~y, we generate 2 different 4 bit code tables based on the two initialisation vectors d" = [1 3 7l and db = t1 2 3] and this time we show the metric d2(m~
~mse). The kernel is chosen to 111. See Table 3.
CA 022380l5 l998-05-l5 Level Code d2lm) Code d2(m) m da db 1Ei 0000 190 0001 195 Table 3.
As a comparison a straigth code has d2~m) = 85 for each code word.
7.3 Variations of code tables.
The bit-columns of the code tables can be changed in any order without changing the metric (distance~.
Ones can be exchanged by zeros and vice versa without changing the metric .
CA 0223801~ 1998-0~
Baang-~ code.
The Baang-1 code gives the minimum distance (or mse) to code words with Hamming distance one, i.e. for code words with single bit errors, to the kernel.
Alternative code words on each side of the kernel are designed in such a way that the levels associated with the Hamming distance = 1 to the code word considered are minimised.
The kernel has the least possible distance (or mse) and the distance measure (or mse) increases further away from the kernel.
The Baang-1 code is particularly useful for representing information that has a peak in its pdf.
Although the design method has been described for a short 3-bit code, for the purpose of illustration, the same design principle is applicable for longer codes. Furthermore, the maximum gain compared to a straight code increases with the length of the code words.
Baang-2 code.
With this design method we have a possibility to adapt the error distances to the peak(s) of the pdf in such a way that small error distances are allocated to the peak(s) of the pdf. The way to control this allocation is by means of the initialisation process i.e. how we set the reference code words related to the kernel - the distances d1, d2, d3, ... dk, where 0 < k < n-1 and where each d-component takes on distinct integer values in the interval [1, 2(n~ 1]. The d-values are put at positions where the pdf has high values.
CA 0223801~ 1998-OS-lS
Columns in the code table can be reordered in any order without changing the distance measures. Ones can be replaced by zeros and vice versa without changing the distance measures. The initialisation process can, of course, also generate new code words downwards (instead of upwards) and the code table can be generated in an apparent way based on the steps described above ( a symmetry principle).
The arrangement in accordance with the invention comprises means for allocating a new code word to the other side of the kernel in cases where the end of the code table has been reached on the side where said code word should be allocated according to previous steps. The arrangement also comprises means to stop new code word generation when all 2" code words have been generated and means for removing not allowed code word(s), which with a suitable choice of the kernel as previously described, is (are) allocated towards the end(s) of the code table. A number of variations of the above mentioned embodiments are of course possible Bit-columns in the code table can e.g. be reordered in any order without changing the distance measures. Binary "ones" can be replaced by "zeros"
and vice versa without changing the distance measures.
7.1 .1 ) Initialise the code table.
7.1.1 .a) Pick the kernel, for example 111, and put it on that level position that corresponds to the peak of the level pdf, in our case, say level 3.
Level Code Word 3 111 (kernel) 7.1.1.b) On either side of the kernel, insert two new code words with the Hamming distance one to the kernel. In this case we change the leftmost bit first and then the next bit in order. However, any order of bitchange can be used .
Level Code Word Distance d1 3 111 1 +1 +?=?
7.1.1.c) The third bit, the rightmost bit of the kernel, is now changed and the new code word is put on level 1 (or level 5) Level Code Word Distance d1 d2 3 111 1+1+2=412+12+22=6 CA 0223801~ 1998-0~
With this initialisation we have packed all code words with the Hamming distance one as close as possible ~in the ~uclidean distance sense) to the kernel .
Should there be more than 3 bits in the code words, say n bits, the same procedure is applied and the result is an initialisation of (n + 1 ) code words.
7.1.2) Continue to build the table.
7.1 .2.a) The next code word to consider is the one closest to the kernel and with the maximum degree of freedom to design new code words in a compact way.
Level 2 with the code word 011 has code words on each side, but level 4 with the code word 101 has a code word only on one side (111), and thus has the maximum degree of freedom. This code word (101) is called the 1 :st reference.
Thus, changing the bits in the 1 :st reference we get the following table:
(here we use the same order for changing bits as was used in the initialisation, but any other order can be used as well) Level Code Word Distance d1 d2 3 111 1 +1+2--4 6 4 --> 101 1 t1 tZ=4 6 CA 02238015 1998-0~-15 W O ~712Q395 10 PCT~SE96/01555 If not the bit changed code word is in the table we add it as close as possible to the reference word.
7.1.2.b) Following the principle in 2a) we now consider level 2 and choose code word 011 as the 2:nd reference and get Level Code Word Distance dl d2 2 --> 011 1+3+2--6 14 3 111 1+1 +2=4 6 4 101 1 + 1+2=4 6 7.1 .2.c) For the next reference to consider (110 or 001) there is no difference in the distance to the kernel and the number of freedom is the same and so 2a) does not give us the information how to proceed and any side can be used.
In this example we stick to the symmetry principle and process that code word that is on the opposite side of the kernel of the last processed code word.
In this case it means level 6 with the reference word 001, and the result is Level Code Word Distance d1 d2 2 011 1 +3+2z614 3 111 1+1+234 6 4 101 1+ 1+2--4 6 --> 001 1+3+2=6 14 CA 0223801~ 1998-0~
7.1.3) All 8 code words are now associated with a level and we just calculate the distance d1 (m) and mse d2(m) for the remaining 4 code words.
~, Table 2 shows the final result, the 3 bit Baang-1 code.
Straight Code Baang-1 Code Code Distance Code Distance dl d1 d2 o 000 4+2+1 =7 010 1 +7+2=10 54 001 4+2+1=7 110 1+5+2=8 30 2 010 4+2+1=7 011 1+3+2=6 14 3 011 4+2+1=7 111 1+1+2=4 6 4 100 4+2+1 =7 101 1 +1 +2-4 6 101 4+2+1 =7 001 1 +3+2=6 14 6 110 4+2+1=7 100 1+5+2=8 30 7 111 4+2+1=7 000 1+7+2=10 54 Table 2.
As can be seen from the table, the distance d2 for the straight code is 21 for each code word.
7.1.3) Te~ .alion of the table.
If a new code word designed from a reference and allocated according to the principles outlined above falls outside one of the end levels, a reference word from the other side of the kernel is used. This may happen, for example, when the peak of the pdf is not in the central part of the level distribution. The following example illustrates the principle, where we assume that the peak of the pdf is located at level 5.
CA 022380l5 l998-05-l5 Step 1 Step 2 Step 3 Step 4 Level Code Code Code CodeDistance d1 0 0001 +1+4=6 1 100 1001 +6+5=12 2 010 010 0105+2+2=9 3 001 001 -->0013+1+3=7 4 --> 011 011 0111 ~1+2=4 111 111 111 1111 +1+2=4 6 101 --> 101 1013~1+5=9 7 110 110 110~;+6+2=13 7.2 More general pdf (Basng-2 code).
This technique may be used when the pdf has a more general shape and the design steps are as follows. Here we illustrate the method by a 4 bit code (n = 4) .
7.2.1) Initialise the code tsble.
7.2.1.a) Pick a (n-1) bit code word, for example 111, denoted kernel, and put it on the level position that corresponds to the peak of the level pdf, say level 8.
Level Code Word 7.2.1.b) Insert a maximum of ~n-1) new code words with Hamming distance one from the kernel and put them on the distances d = ld1, d2, ... dk ~ units from the kernel, in our example we choose dt = 1, d2 = 3 and d3 = 5 and we get, -CA 022380l5 l998-05-l5 WO 97/20395 PCT/SE96/015~5 Level Code Word __ 5 101 d3 6 d2 ~~~~i~ 7 110 -- 8 111 (kernel~
~0 7.2.Z) Continue to build the table.
7.2.2.a) The code table is built from the kernel and upwards and the code word associated with the distance d1 is used as the 1 :st reference, i.e. 110.
(The index in the distance measure is taken as the order in which the references are processed).
Changing, the bits in the 1 :st reference and packing the new words as close as possible to the kernel we get Level Code Word 7 --> 110 Next we use the code word associated with d2, 101, as a reference, 3 0 changing the bits and obtain W O 97/20395 14 PCT/S~96101555 Level Code Word 5 -> 101 Next reference corresponding to d3, 01 1, does not give any new code word, and all the references have now been used.
~5 2.2.2~b) With the code word next to the kernel as a reference, 1 10, chan~e the bits and fili up the table with new code words in positions as close as possible to the kernel. In this case 1 10 does not give any new code word and so continue with the next code word counted from the kernel, namely 010.
Level Code Word 6 --~ 010 and we have all 8 code words.
CA 0223801~ 1998-0~
7.2.2.c) To obtain the 16 possible code words for the 4 bit code we take the mirror picture of the 8 code words below the 8:th word (or above the 1 :st word) and put a 1 (or 0) at the leftmost position (or rightmost position or any position) of the 8 original words and a 0 (or 1 ) at the same bit position in the reflected 8 words,.i.e. one implementation gives Level Code WordDistance d1 The design method ensures a symmetrical distance distribution d1 ~m) and d2(m) and it allows a flexibility to adapt the distance distribution to the actual pdf of the level in such a way that a small distance is allocated to the peak(s) of the pdf, thereby making the mean error distance per bit ~l and the squared distance per bit ~2 small.
, 30 Forbidden code words are removed from the table.
CA 022380l5 l998-05-l5 To illustrate the flexibility of the Baang-2 code methodolo~y, we generate 2 different 4 bit code tables based on the two initialisation vectors d" = [1 3 7l and db = t1 2 3] and this time we show the metric d2(m~
~mse). The kernel is chosen to 111. See Table 3.
CA 022380l5 l998-05-l5 Level Code d2lm) Code d2(m) m da db 1Ei 0000 190 0001 195 Table 3.
As a comparison a straigth code has d2~m) = 85 for each code word.
7.3 Variations of code tables.
The bit-columns of the code tables can be changed in any order without changing the metric (distance~.
Ones can be exchanged by zeros and vice versa without changing the metric .
CA 0223801~ 1998-0~
Baang-~ code.
The Baang-1 code gives the minimum distance (or mse) to code words with Hamming distance one, i.e. for code words with single bit errors, to the kernel.
Alternative code words on each side of the kernel are designed in such a way that the levels associated with the Hamming distance = 1 to the code word considered are minimised.
The kernel has the least possible distance (or mse) and the distance measure (or mse) increases further away from the kernel.
The Baang-1 code is particularly useful for representing information that has a peak in its pdf.
Although the design method has been described for a short 3-bit code, for the purpose of illustration, the same design principle is applicable for longer codes. Furthermore, the maximum gain compared to a straight code increases with the length of the code words.
Baang-2 code.
With this design method we have a possibility to adapt the error distances to the peak(s) of the pdf in such a way that small error distances are allocated to the peak(s) of the pdf. The way to control this allocation is by means of the initialisation process i.e. how we set the reference code words related to the kernel - the distances d1, d2, d3, ... dk, where 0 < k < n-1 and where each d-component takes on distinct integer values in the interval [1, 2(n~ 1]. The d-values are put at positions where the pdf has high values.
CA 0223801~ 1998-OS-lS
Columns in the code table can be reordered in any order without changing the distance measures. Ones can be replaced by zeros and vice versa without changing the distance measures. The initialisation process can, of course, also generate new code words downwards (instead of upwards) and the code table can be generated in an apparent way based on the steps described above ( a symmetry principle).
The arrangement in accordance with the invention comprises means for allocating a new code word to the other side of the kernel in cases where the end of the code table has been reached on the side where said code word should be allocated according to previous steps. The arrangement also comprises means to stop new code word generation when all 2" code words have been generated and means for removing not allowed code word(s), which with a suitable choice of the kernel as previously described, is (are) allocated towards the end(s) of the code table. A number of variations of the above mentioned embodiments are of course possible Bit-columns in the code table can e.g. be reordered in any order without changing the distance measures. Binary "ones" can be replaced by "zeros"
and vice versa without changing the distance measures.
Claims (14)
1. Method in a telecommunication system for making robust fixed length codes, which system comprises an input device (101) containing an information source that generates a variable that is a function of time F(t), wherein a probability density function for values occurring in said function of time F(t) has a maximum and which system comprises a digitiser (102) and an encoder (103) for digitising and encoding said values into code words such that the Hamming distance between those among said code words that are adjacent to one another is lower for those among said values that are near the maximum of the probability density function than for values that are further away from said maximum.
2. Method in a telecommunications system according to claim 1 wherein when encoding said values of said function of time F(t) a fixed length code is used designed according to the following steps:
- initialising of a code table, wherein a code word, hereinafter called the kernel, is allocated to a value the position of which corresponds to a peak of a probability density function of the transmitted information and wherein said kernel can be chosen arbitrarily as a n-bit code word;
- generating n new code words with the Hamming distance one to said kernel;
- packing said new code words as close as possible on either side of the kernel in order to build said code table.
- initialising of a code table, wherein a code word, hereinafter called the kernel, is allocated to a value the position of which corresponds to a peak of a probability density function of the transmitted information and wherein said kernel can be chosen arbitrarily as a n-bit code word;
- generating n new code words with the Hamming distance one to said kernel;
- packing said new code words as close as possible on either side of the kernel in order to build said code table.
3. The method for claim 2, comprising the following further steps:
- continuing to build the table, where a first code word to consider, hereafter called the first reference word, is the code word closest to the kernel that either, when n is odd, is located on that side of the kernel that has the least number of initialised code words or, when n is even, is located on any side of the kernel;
- generating n new code words with Hamming distance one to said first reference word and packing said n new code words, if not already in the table, as close as possible to the said first reference word.
- continuing to build the table, where a first code word to consider, hereafter called the first reference word, is the code word closest to the kernel that either, when n is odd, is located on that side of the kernel that has the least number of initialised code words or, when n is even, is located on any side of the kernel;
- generating n new code words with Hamming distance one to said first reference word and packing said n new code words, if not already in the table, as close as possible to the said first reference word.
4. The method of claim 3, comprising the following further steps:
- selecting a second reference word from the not yet selected side of the kernel and as close as possible to the kernel;
- generating n new code words with Hamming distance one to said second reference word and packing said n new code words, if not already in the table, as close as possible to the said second reference word.
- selecting a second reference word from the not yet selected side of the kernel and as close as possible to the kernel;
- generating n new code words with Hamming distance one to said second reference word and packing said n new code words, if not already in the table, as close as possible to the said second reference word.
5. The method of claim 4, comprising the following further steps:
- repetitively selecting new reference words as close as possible to the kernel in an alternating order and increasing distance from the kernel;
- generating n new code words with Hamming distance one to current reference word and packing said n new code words, if not already in the table, as close as possible to the said current reference word.
- repetitively selecting new reference words as close as possible to the kernel in an alternating order and increasing distance from the kernel;
- generating n new code words with Hamming distance one to current reference word and packing said n new code words, if not already in the table, as close as possible to the said current reference word.
6. In a telecommunications system that comprises transmission of information over a channel a method to encode the transmitted information wherein the fixed length code is designed according to the following steps, initialising of a code table, wherein a code word, hereafter called the kernel, is allocated to a position corresponding to the peak of the probability density function of the transmitted information and wherein said kernel can be chosen arbitrarily as a n-bit code word but in cases where there are less than 2n allowed code words in the table the kernel can be favourably selected as the code word with the largest Hamming distance to the not allowed code word(s), generating n new code words with the Hamming distance one to said kernel and packing said new code words as close as possible on either side of the kernel.
7. The method of claim 6 further comprising the following steps continuing to build the table, where the first code word to consider, hereafter called the first reference word, is the code word closest to the kernel that either, when n is odd, is located on that side of the kernel that has the least number of initialised code words or, when n is even, is located on any side of the kernel, generating n new code words with Hamming distance one to said first reference word and packing said n new code words, if not already in the table, as close as possible to the said first reference word.
8. The method of claim 7, comprising the following further steps:
- selecting a second reference word from the not yet selected side of the kernel and as close as possible to the kernel;
- generating n new code words with Hamming distance one to said second reference word and packing said n new code words, if not already in the table, as close as possible to the said second reference word.
- selecting a second reference word from the not yet selected side of the kernel and as close as possible to the kernel;
- generating n new code words with Hamming distance one to said second reference word and packing said n new code words, if not already in the table, as close as possible to the said second reference word.
9. The method of claim 8, comprising the following further steps:
- repetitively selecting new reference words as close as possible to the kernel in an alternating order and increasing distance from the kernel;
- generating n new code words with Hamming distance one to current reference word and packing said n new code words, if not already in the table, as close as possible to the said current reference word.
- repetitively selecting new reference words as close as possible to the kernel in an alternating order and increasing distance from the kernel;
- generating n new code words with Hamming distance one to current reference word and packing said n new code words, if not already in the table, as close as possible to the said current reference word.
10. In a telecommunications system that comprises transmission of information over a radio channel a method to encode the transmitted information wherein the fixed length code is designed according to the steps described in claim 1.
11. In a telecommunications system that comprises transmission of video information over a radio channel a method to encode the transmitted video information wherein the fixed length code is designed according to the steps described in claim 1.
12. In a telecommunications system that comprises transmission of information over a channel a method for making robust fixed length codes, which system comprises an input device (101) containing an information source that generates a variable that is a function of time F(t), wherein a probability density function for values occurring in said function of time F(t) has at least one maximum and which system comprises a digitiser (102) and an encoder (103) for digitising and encoding said values into code words such that the Hamming distance between those among said code words that are adjacent to one another is lower for those among said values that are near the at least one maximum of the probability density function than for values that are further away from said at least one maximum wherein the fixed length code is designed according to the following steps:
- initiating a code table, wherein a code word, hereafter called the kernel, is allocated to a position corresponding to an estimated or real symmetry line of the probability density function of the transmitted information, said kernel can be chosen arbitrarily as a (n-1) bit word, where n is the number of bits in a code word, but in cases where there are less than 2 n allowed code words in the table, the kernel can be favourably selected as the (n-1) bit word with the largest Hamming distance to (n-1) bits of the not allowed code word(s);
- generating k new words, where 1 ~ k ~ (n-1), with the Hamming distance one to said kernel and said new words are put at level distances d =[d1, d2 ... dk ] units from the kernel, and each said level distance is distinct from any other level distance but can be of any integer value in the level range [1, 2(n-1)-1] units, and where the said level distance index determines the order in which the corresponding words are processed, and by selecting the said level distances d = [d1, d2 ... dk] affect the error distances d1 (m) and d2(m) to match the probability density function in such a way that, choosing d = [1, 2, ..., (n-1)] units, the minimum error distances d1(m) and d2(m) are allocated towards the kernel with a peak at the kernel, and choosing d = [(n-1), (n-2), ..., 1] units, the minimum error distances d1(m) and d2(m) have a wider distribution compared to previous case, and choosing d = x where x is a vector of k distinct elements in the range [1, 2(n-1)-1] units, with the design flexibility to move the minimum error distances d1(m) and d2(m) away from the kernel and where the minimum error distances occur in the region where the elements in x are compactly positioned;
- continuing to build the table, where the first word to consider is the one pointed to by d1, hereafter called the first reference word;
- generating (n-1) new words with Hamming distance one to said first reference word and packing said (n-1) new words, if not already in the table, as close as possible to the kernel;
- selecting a second reference word pointed to by d2;
- generating (n-1) new words with Hamming distance one to said second reference word and packing said (n-1) new words, if not already in the table, as close as possible to the kernel;
- repetitively selecting new reference words pointed to by successive elements in d, until the last element dk is reached, and;
- generating (n-1) new words with Hamming distance one to current reference word and packing said (n-1) new words, if not already in the table, as close as possible to the kernel;
- completing the table in cases the table is not full (with 2(n-1) words) after said initialisation process, by which repetitively the word closest to the kernel is taken as a reference word and means for generating (n-1) new words with Hamming distance one to current reference word packing said (n-1) new words, if not already in the table, as close as possible to the kernel;
- stopping new word generation when 2(n-1) words have been generated;
- generating 2(n-1) words by flipping the existing 2(n-1) words up-downbelow the kernel;
- putting a one (or zero) in any column position in the first 2(n-1) words and a zero (or one) in the same column position in the next 2(n-1) words, whereby a code table with 2n code words is generated, and;
- removing not allowed code word(s), which with a suitable choice of the kernel as previously described, is (are) allocated towards the end(s) of the code table.
- initiating a code table, wherein a code word, hereafter called the kernel, is allocated to a position corresponding to an estimated or real symmetry line of the probability density function of the transmitted information, said kernel can be chosen arbitrarily as a (n-1) bit word, where n is the number of bits in a code word, but in cases where there are less than 2 n allowed code words in the table, the kernel can be favourably selected as the (n-1) bit word with the largest Hamming distance to (n-1) bits of the not allowed code word(s);
- generating k new words, where 1 ~ k ~ (n-1), with the Hamming distance one to said kernel and said new words are put at level distances d =[d1, d2 ... dk ] units from the kernel, and each said level distance is distinct from any other level distance but can be of any integer value in the level range [1, 2(n-1)-1] units, and where the said level distance index determines the order in which the corresponding words are processed, and by selecting the said level distances d = [d1, d2 ... dk] affect the error distances d1 (m) and d2(m) to match the probability density function in such a way that, choosing d = [1, 2, ..., (n-1)] units, the minimum error distances d1(m) and d2(m) are allocated towards the kernel with a peak at the kernel, and choosing d = [(n-1), (n-2), ..., 1] units, the minimum error distances d1(m) and d2(m) have a wider distribution compared to previous case, and choosing d = x where x is a vector of k distinct elements in the range [1, 2(n-1)-1] units, with the design flexibility to move the minimum error distances d1(m) and d2(m) away from the kernel and where the minimum error distances occur in the region where the elements in x are compactly positioned;
- continuing to build the table, where the first word to consider is the one pointed to by d1, hereafter called the first reference word;
- generating (n-1) new words with Hamming distance one to said first reference word and packing said (n-1) new words, if not already in the table, as close as possible to the kernel;
- selecting a second reference word pointed to by d2;
- generating (n-1) new words with Hamming distance one to said second reference word and packing said (n-1) new words, if not already in the table, as close as possible to the kernel;
- repetitively selecting new reference words pointed to by successive elements in d, until the last element dk is reached, and;
- generating (n-1) new words with Hamming distance one to current reference word and packing said (n-1) new words, if not already in the table, as close as possible to the kernel;
- completing the table in cases the table is not full (with 2(n-1) words) after said initialisation process, by which repetitively the word closest to the kernel is taken as a reference word and means for generating (n-1) new words with Hamming distance one to current reference word packing said (n-1) new words, if not already in the table, as close as possible to the kernel;
- stopping new word generation when 2(n-1) words have been generated;
- generating 2(n-1) words by flipping the existing 2(n-1) words up-downbelow the kernel;
- putting a one (or zero) in any column position in the first 2(n-1) words and a zero (or one) in the same column position in the next 2(n-1) words, whereby a code table with 2n code words is generated, and;
- removing not allowed code word(s), which with a suitable choice of the kernel as previously described, is (are) allocated towards the end(s) of the code table.
13. In a telecommunications system that comprises transmission of information over a radio channel a method to encode the transmitted information wherein the fixed length code is designed according to the steps described in 6.
14. In a telecommunications system that comprises transmission of video information over a radio channel a method to encode the transmitted video information wherein the fixed length code is designed according to the steps described in 6.
Applications Claiming Priority (5)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| US761495P | 1995-11-29 | 1995-11-29 | |
| SE9504270-1 | 1995-11-29 | ||
| SE9504270A SE9504270D0 (en) | 1995-11-29 | 1995-11-29 | A method and an arrangement related to communication systems |
| US60/007,614 | 1995-11-29 | ||
| PCT/SE1996/001555 WO1997020395A1 (en) | 1995-11-29 | 1996-11-27 | A method and an arrangement for making fixed length codes |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA2238015A1 true CA2238015A1 (en) | 1997-06-05 |
Family
ID=29407549
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA 2238015 Abandoned CA2238015A1 (en) | 1995-11-29 | 1996-11-27 | A method and an arrangement for making fixed length codes |
Country Status (1)
| Country | Link |
|---|---|
| CA (1) | CA2238015A1 (en) |
-
1996
- 1996-11-27 CA CA 2238015 patent/CA2238015A1/en not_active Abandoned
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