CN101753147B - Iterative decoding algorithm of quadratic residue code - Google Patents

Abstract

The invention discloses an iterative decoding algorithm of a quadratic residue code, which is suitable for decoding all quadratic residue codes. The decoding algorithm comprises the steps of: obtaining a plurality of known characteristic values for providing digital signals; calculateing a plurality of unknown characteristic values with the plurality of known characteristic values; calculating to obtain an error position polynome with a no anti-root bailey root algorithm; and calculating the error position of the error position polynome with a chien search algorithm to be capable of modifying the relative binary digit of the error digital signals to obtain a correct code word. The algorithms can calculate the error position polynome to improve the capability for decoding the quadratic residue code.

Description

The decoding algorithm of quadratic residue code

Technical field

The present invention relates to a kind of decoding algorithm, relate in particular to a kind of decoding algorithm of applicable all quadratic residue codes.

Background technology

In digital Age, the signal such as sound, image all adopts digitlization to process such as, such as what's frequently heard can be repeated in detail Digital Television, bluetooth earphone, DVD audio-video optical disk and WAP mobile communication etc.Reach high-quality phonotape and videotape when making the digital product signal can effectively read with long-distance transmissions and reappear, generally signal is done the processing arrangement of encoding and decoding.

Quadratic residue code (Quadratic Residue Code) is widely used in the digital encoding and decoding of every field at present, its decoding algebraic decoding modes that adopt are eliminated the middle unknown characteristics value (Unknow Syndrome) of Newton's identities (Newton ' s identities) in order to obtain the coefficient of error location polynomial (ErrorPolynomial) more, further obtain error location polynomial (Error Polynomial).But along with the quadratic residue code length increases, high order equation more difficult solution that finds in limited body of using the algebraic decoding mode to produce, thereby so that error location polynomial is difficult to acquisition.

The shortcoming that when the quadratic residue code length increases, is difficult to obtain error location polynomial for solving the algebraic decoding mode, generally adopt without anti-Gen Bailigen (Inverse-free Berlekamp Massey) algorithm to calculate error location polynomial, but quadratic residue code there is no enough continuously known features value (KnowSyndrome) as inputting without anti-Gen Bailigen algorithm, to calculate correct and wrong position multinomial, to reach being correctly decoded in the quadratic residue code error correcting capability, therefore, how to calculate error location polynomial and become important problem.

Summary of the invention

In view of this, for addressing the above problem, the present invention proposes the decoding algorithm of applicable all quadratic residue codes.

The invention provides the decoding algorithm of quadratic residue code, utilize the digital signal that provides to calculate first a plurality of known features values (Known Syndrome), under mistake occurs, set wrong number and the error correcting capability value of occuring, after then utilizing a plurality of known features values to calculate a plurality of unknown characteristics values (UnknownSyndrome), a plurality of known features values and a plurality of unknown characteristics value are calculated error location polynomial with one without anti-Gen Bailigen (Inverse-free Berlekamp Massey) algorithm as known.

At this moment, whether multinomial the highest power in misjudgment position is consistent with mistake generation number.If judged result is consistent, then calculates this error location polynomial with chien search (Chien-search) algorithm and find out its solution root and separate radical.If judged result is inconsistent namely represent the error location polynomial of trying to achieve non-correct, then mistake is occured that number increases certain value and whether decision error generation number exceeds the error correcting capability value.If decision error generation number then recomputates a plurality of unknown characteristics values and returns without anti-Gen Bailigen algorithm and again look for one group of error polynomial without exceeding the error correcting capability value.If exceeding the error correcting capability value, decision error generation number then finishes the decoding algorithm program.

Calculate the Xie Genyu solution radical of error location polynomial with the chien search algorithm after, further whether the polynomial solution root in misjudgment position is consistent with mistake generation number.If be judged as when no, then get back to and above-mentionedly wrongly occur that number increases certain value and whether decision error generation number exceeds in the error correcting capability value decoding program.If be judged as when being, then seek the polynomial wrong occurrence positions in position that makes mistake, revise the relative bit of digital signal mistake occurrence positions, obtain whereby correct code word.

Can calculate error location polynomial by quadratic residue code decoding algorithm of the present invention, and applicable all quadratic residue codes, and via proposing the decoding capability that the existing quadratic residue code of decoding algorithm lifting can't be reached after the actual computer checking.

Description of drawings

Fig. 1 is decoding algorithm decoding quadratic residue code (89,45,17) the embodiment flow chart of quadratic residue code of the present invention;

Fig. 2 is decoding algorithm decoding quadratic residue code (71,36,11) the embodiment flow chart of quadratic residue code of the present invention; And

Fig. 3 is decoding algorithm decoding quadratic residue code (79,40,15) the embodiment flow chart of quadratic residue code of the present invention.

Wherein, description of reference numerals is as follows:

Step S100 provides digital signal, utilizes digital signal to calculate a plurality of known features values, if wrong number and the error correcting capability value of occuring set in wrong generation

Step S110 utilizes a plurality of known features values to calculate a plurality of unknown characteristics values of quadratic residue code (89,45,17)

Step S120 utilizes a plurality of known features values and a plurality of unknown characteristics value as known, via calculating error location polynomial without anti-Gen Bailigen algorithm

Whether step S130 misjudgment position the highest polynomial power is consistent with mistake generation number

The Xie Genyu that step S140 calculates error location polynomial separates radical

Step S150 mistake generation number increases certain value and whether decision error generation number exceeds the error correcting capability value

Whether the polynomial solution radical in step S160 misjudgment position is consistent with mistake generation number

Step S170 seeks the make mistake polynomial wrong occurrence positions in position and numerical value corresponding to the occurrence positions that corrects mistakes

Step S200 provides digital signal, utilizes digital signal to calculate a plurality of known features values, if during wrong generation, sets wrong number and the error correcting capability value of occuring

Step S210 utilizes a plurality of known features values to calculate a plurality of unknown characteristics values of quadratic residue code (71,36,11)

Step S220 utilizes a plurality of known features values and a plurality of unknown characteristics value as known, via calculating error location polynomial without anti-Gen Bailigen algorithm

Whether step S230 misjudgment position the highest polynomial power is consistent with mistake generation number

The Xie Genyu that step S240 calculates error location polynomial separates radical

Step S250 mistake generation number increases certain value and whether decision error generation number exceeds the error correcting capability value

Whether the polynomial solution radical in step S260 misjudgment position is consistent with mistake generation number

Step S270 seeks the make mistake polynomial wrong occurrence positions in position and numerical value corresponding to the occurrence positions that corrects mistakes

Step S300 provides digital signal, utilizes digital signal to calculate a plurality of known features values, if during wrong generation, sets wrong number and the error correcting capability value of occuring

Step S310 utilizes a plurality of known features values to calculate a plurality of unknown characteristics values of quadratic residue code (79,40,15)

Step S320 utilizes a plurality of known features values and a plurality of unknown characteristics value as known, via calculating error location polynomial without anti-Gen Bailigen algorithm

Whether step S330 misjudgment position the highest polynomial power is consistent with mistake generation number

The Xie Genyu that step S340 calculates error location polynomial separates radical

Step S350 mistake generation number increases certain value and whether decision error generation number exceeds the error correcting capability value

Whether the polynomial solution radical in step S360 misjudgment position is consistent with mistake generation number

Step S370 seeks the make mistake polynomial wrong occurrence positions in position and numerical value corresponding to the occurrence positions that corrects mistakes

Embodiment

For making having further understanding in the object of the invention, the algorithm, the spy is described in detail as follows in conjunction with related embodiment and accompanying drawing:

Please also refer to Fig. 1, it is the decoding algorithm of quadratic residue code of the present invention (Quadratic Residue Code), decoding quadratic residue code (89,45,17) embodiment, and decoding process includes the following step:

Digital signal is provided, utilizes digital signal to calculate a plurality of known features values (KnownSyndrome), if during wrong generation, set wrong number and the error correcting capability value of occuring, such as step S100.In this step, digital signal can calculate a plurality of known features values via equation 1, and wherein wrong number occurs is parameter v to hypothesis, and initial setting up numerical value is 1, and the error correcting capability value of quadratic residue code (89,45,17) is 8.

S _i=r (β ⁱ)=c (β ⁱ)+e (β ⁱ)=e (β ⁱ)=e ₀+ e ₁β+... + e ₈₈β ⁸⁸(equation 1)

S in the equation 1 wherein _iBe the known features value, β is GF (2 ¹¹) primary n root (Primitiventh Root of Unity) in the body, n is code length.

Utilize a plurality of known features values to calculate a plurality of unknown characteristics values (Unknown Syndrome) of quadratic residue code (89,45,17), such as step S110.At first must make set I={i ₁, i ₂..., i _V+1And set J={j ₁, j ₂..., j _V+1Be 0,1 ..., the subclass of n-1}, n are code length.And the definition large minor matrix X of (v+1) * v (I) and X (J), matrix notation is as follows:

Further define matrix S (I, J)=X (I) X (J) ^TWith its theorem, matrix X (J) wherein ^TBe expressed as the transposed matrix of X (J), S (I, J) then is the matrix of a size former (v+1) * (v+1).

Theorem 1. matrix S (I, J) have following pattern

Wherein S lower is designated as the remainder except rear gained by n, and n is code length.And the determinant of matrix S (I, J) equals zero, namely: det (S (I, J))=0.

If matrix S (I, J) contains a plurality of unknown characteristics values, then det (S (I, J))=0 can regard the equation of a unknown characteristics value as, and coefficient is comprised of the known features value.If only have a unknown characteristics value in the matrix S (I, J)

So

Can be obtained by following theorem, wherein

Mean the unknown characteristics value under v wrong generation of hypothesis is several.

If only have a unknown characteristics value among theorem 2. matrix S (I, J)

Then

Can be expressed as a fraction, the value of this fraction is to belong to matrix S (I, J) wherein to be divided by after two determinants of a matrix expansion.If

Appear at the capable j row of i in the matrix S (I, J), then $S_r^{\left(v\right)}=\frac{\det \left({\mathrm{\Δ}}_0\right)}{\det \left(\mathrm{\Δ}\right)},$ Δ wherein ₀Refer in (v+1) * (v+1) matrix S (I, J)

The address capable j of i at place is listed as by zero and replaces, and Δ refers to eliminate in the matrix S (I, J) submatrix that the capable and j of i is listed as formed v * v.

Can be calculated as follows mode via the unknown characteristics value of 2 residue codes of theorems (89,45,17) calculates and learns:

Case0: if when inerrancy occurs, S3 (0)=0; S ₁₃ ⁽⁰⁾=0.

Case1：S ₃( ¹)＝S ₁ ³；S ₁₃ ⁽¹⁾＝S ₁ ¹³。

Following Case v a) and b) define respectively I and J set.Recycling theorem 1 further just can calculate two unknown characteristics value S with theorem 2 ₃With S ₁₃

Case2：

a)I ₂＝{0，1，2}，J ₂＝{0，1，87}。

$S(I_2,J_2)=\left[\begin{array}{ccc}{S}_0& S_1& S_2\\ S_1& S_2& S_3^{\left(2\right)}\\ S_{87}& S_{88}& S_0\end{array}\right].$

S ₃ ⁽²⁾＝(S ₁S ₂S ₈₈+S ₄S ₈₇)/(S ₁+S ₈₇)。

b)I ₂＝{0，11，88}，J ₂＝{0，2，5}。

$S(I_2,J_2)=\left[\begin{array}{ccc}{S}_0& S_{11}& S_{88}\\ S_2& S_{13}^{\left(2\right)}& S_1\\ S_5& S_{16}& S_4\end{array}\right].$

S ₁₃ ⁽²⁾＝(S ₂S ₁₆S ₈₈+S ₂S ₄S ₁₁+S ₁S ₅S ₁₁)/(S ₅+S ₈₈)。

Case3：

a)I ₃＝{0，1，87，88}，J ₃＝{0，1，2，10}。

$S(I_3,J_3)=\left[\begin{array}{cccc}{S}_0& S_1& S_{87}& S_{88}\\ S_1& S_2& S_{88}& S_0\\ S_2& S_3^{\left(3\right)}& S_0& S_1\\ S_{10}& S_{11}& S_8& S_9\end{array}\right].$

b)I ₃＝{8，9，10，13}，J ₃＝{0，8，59，71}。

$S(I_3,J_3)=\left[\begin{array}{cccc}{S}_8& S_9& S_{10}& S_{13}^{\left(3\right)}\\ S_{16}& S_{17}& S_{18}& S_{21}\\ S_{67}& S_{68}& S_{69}& S_{72}\\ S_{79}& S_{80}& S_{81}& S_{84}\end{array}\right]$

Case4：

a)I ₄＝{1，2，3，10，32}，J ₄＝{0，7，8，77，78}。

b)I ₄＝{0，1，2，87，88}，J ₄＝{0，1，2，3，11}。

Case5：

a)I ₅ ¹＝{1，16，17，20，39，57}，J ₅ ¹＝{0，33，52，68，71，72}

I ₅ ²＝{0，1，8，39，49，72}，J ₅ ²＝{0，1，8，39，49，84}。

In the situation of 5 mistakes, can't only just can define the unknown characteristics value S of all 5 mistakes with one group (I, J) set ₃ ⁽⁵⁾Need to find two groups (I, J) set can contain all 5 wrong corresponding unknown characteristics value S through the simulation test of computer ₃ ⁽⁵⁾

b)I ₅＝{0，1，2，3，4，5}，J ₅＝{0，1，2，8，87，88}。

With the S that obtains in a) ₃ ⁽⁵⁾Be considered as utilizing theorem 1 and theorem 2 further just can calculate two unknown characteristics value S after its characteristic value ₁₃ ⁽⁵⁾

Case6：

a)I ₆ ¹＝{1，2，3，4，9，71，87}，J ₆ ¹＝{0，1，2，7，8，86，87}

I ₆ ²＝{1，2，3，4，5，11，87}，J ₆ ²＝{0，1，5，6，7，86，87}

I ₆ ³＝{1，2，3，4，9，87，88}，J ₆ ³＝{0，1，2，7，8，46，86}。

In the situation of 6 mistakes, can't only just can define the unknown characteristics value S of all 6 mistakes with one group (I, J) set ₃ ⁽⁶⁾Need to find three groups (I, J) set through the simulation test of computer.The determinantal expansion of this three groups (I, J) set all is nonlinear S ₃ ⁽⁶⁾Multinomial knows that by theorem 2 these determinants equal zero.In other words, these three multinomials can have identical liner factor.Therefore, obtain these three polynomial liner factors and namely can define unique S ₃ ⁽⁶⁾Value.At b) in, can be with S ₃ ⁽⁶⁾Being considered as a known value uses.

b)I ₆ ¹＝{0，1，2，3，4，5，6}，J ₆ ¹＝{0，1，2，3，7，87，88}。

With the S that obtains in a) ₃ ⁽⁶⁾Be considered as utilizing theorem 1 and theorem 2 further just can calculate two unknown characteristics value S after its characteristic value ₁₃ ⁽⁶⁾

Case7：

a)I ₇ ¹＝{2，3，4，6，7，8，14，20}，J ₇ ¹＝{0，2，4，14，65，85，86，87}，I ₇ ²＝{1，2，3，4，6，8，14，87}，J ₇ ²＝{0，2，3，4，6，8，86，87}，I ₇ ³＝{1，2，3，4，9，10，71，87}，J ₇ ³＝{0，1，2，7，8，46，86，87}。

Identical with the situation of 6 mistakes, the necessary unknown characteristics value S that uses 7 mistakes of three groups of (I, J) sets definitions ₃ ⁽⁷⁾At S ₃ ⁽⁷⁾After being defined, b) middle S ₃ ⁽⁷⁾Use with regard to being regarded as a known value.

b)I ₇ ¹＝{0，1，3，6，8，17，40，84}，J ₇ ¹＝{0，4，5，6，8，17，39，47}，I ₇ ²＝{0，1，3，6，8，17，40，88}，J ₇ ²＝{0，4，5，6，8，17，47，88}，I ₇ ³＝{1，2，4，7，9，18，42，87}，J ₇ ³＝{0，4，5，7，38，46，83，87}。

Making them after the determinantal expansion of three groups (I, J) set is zero, can obtain three S ₁₃ ⁽⁷⁾Linear equation.S ₁₃ ⁽⁷⁾Just can be given unique decision by one of them equation.

Case8：

a)I ₈ ¹＝{1，2，3，6，17，20，45，84，87}，J ₈ ¹＝{0，3，4，5，8，19，22，47，86}，I ₈ ²＝{1，3，5，6，7，11，17，39，84}，J ₈ ²＝{0，1，3，5，11，17，39，84，86}，I ₈ ³＝{1，2，3，4，5，6，9，20，87}，J ₈ ³＝{0，1，2，3，5，19，44，86，87}，I ₈ ⁴＝{6，7，8，9，40，53，56，57，72}，J ₈ ⁴＝{0，15，16，38，39，40，41，72，81}。

Identical with the situation of 7 mistakes, the necessary unknown characteristics value S that uses 8 mistakes of four groups of (I, J) sets definitions ₃ ⁽⁸⁾At S ₃ ⁽⁸⁾After being defined, b) middle S ₃ ⁽⁸⁾Use with regard to being regarded as a known value.

b)I ₈ ¹＝{2，6，7，8，9，18，53，85，87}，J ₈ ¹＝{0，2，3，4，14，16，38，82，83}，I ₈ ²＝{0，1，2，3，4，5，6，20，87}，J ₈ ²＝{0，1，2，3，4，5，8，44，87}，I ₈ ³＝{1，2，3，4，5，6，9，20，45}，J ₈ ³＝{0，1，2，3，4，5，19，44，86}。

Making them after the determinantal expansion of three groups (I, J) set is zero, can obtain three S ₁₃ ⁽⁸⁾Linear equation.S ₁₃ ⁽⁸⁾Just can be given unique decision by one of them equation.

Utilize a plurality of known features values and a plurality of unknown characteristics value as known, via without anti-Gen Bailigen (Inverse-free Berlekamp Massey) algorithm mistake in computation position multinomial, such as step S120.In this step, main to utilize a plurality of known features values and a plurality of not known features value that obtain be known input value, further calculates error location polynomial with error location polynomial that can efficient acquisition cyclic code without anti-Gen Bailigen algorithm.Without the following explanation of anti-Gen Bailigen algorithm steps:

Step 1) initial value k=1, r ⁽⁰⁾=1, C ⁽⁰⁾(x)=1, A ⁽⁰⁾And l (x)=1 ⁽⁰⁾=1.

Step 2) calculated difference

${\mathrm{\Δ}}^{\left(k\right)}=\underset{j=1}{\overset{l^{(k-1)}}{\mathrm{\Σ}}}{c}_{j-1}^{(k-1)}{S}_{k-j+1}$

Step 3) mistake in computation position multinomial

C ^(k)(x)＝r ^(k-1)C ^(k-1)(x)-Δ ^(k)A ^(k-1)(x)·x

Step 4) replacement auxiliary variable A ^(k)(x), A ^(k), 1 ^(k)And r ^(k)

Step 5) enter into the next stage: if k=k+1 is k≤2t, then get back to step 2), otherwise stop.

C wherein ^(k)(x) represent the error location polynomial in k stage, Δ ^(k)Be one by known symptom value S _kA difference of calculating, A ^(k)(x), A ^(k), l ^(k)And r ^(k)Be the polynomial auxiliary variable in position that locates errors in the k stage.

Whether misjudgment position the highest polynomial power is consistent with mistake generation number, such as step S130.In this step, utilize and judge whether formula deg (σ (x))=v misjudgment position the highest polynomial power is identical with wrong generation number, wherein, deg (σ (x)) is the highest power of error location polynomial, and v such as aforementioned be the wrong number that occurs.

If misjudgment position the highest polynomial power is identical with the wrong numerical value that number occurs, the Xie Genyu that then calculates error location polynomial separates radical, such as step S140.In this step, the Xie Genyu of error location polynomial separates radical, and both utilize error polynomial to calculate in conjunction with chien search (Chien Search) algorithm.Chien search algorithm described herein, it is used for seeking limited body root of polynomial, and with other algorithm in comparison, the search that applies to the error location polynomial root has better result of calculation.

If number increase certain value when not identical, occurs with mistake in misjudgment position the highest polynomial power and the wrong numerical value that number occurs and whether decision error generation number exceeds the error correcting capability value, such as step S150.This step is that the numerical value with definite value is made as 1, and the error correcting capability value of quadratic residue code (89,45,17) is 8.When mistake generation number exceeds the error correcting capability value, namely can't revise again the digital signal operation, then finish the algorithm program.Otherwise, if when whether wrong generation number did not exceed the error correcting capability value, then a plurality of known features values of Returning utilization calculated a plurality of unknown characteristics values of quadratic residue code (89,45,17), such as step S110.

Whether the polynomial solution radical in misjudgment position is consistent with mistake generation number, such as step S160.First error location polynomial is separated the radical computing, judged that again the solution radical that calculates occurs with wrong whether number is consistent.The employed judgement formula of this step is u=v, and wherein u is the solution radical, and v is the aforesaid wrong number that occurs.

When judging that two numerical value as consistent, seek the polynomial wrong occurrence positions in position that makes mistake, and numerical value corresponding to the occurrence positions that corrects mistakes, such as step S170.In this step, utilize error location polynomial to seek out the digital signal mistake occurrence positions of input, and the numerical value correction that obtains the digital signal errors present is obtained correct value.

Otherwise, if judge two numerical value not when consistent, then return and mistake occured number increases certain value and whether decision error generation number exceeds the error correcting capability value, such as step S150.

Please also refer to Fig. 2, it is the decoding algorithm of quadratic residue code of the present invention, decoding quadratic residue code (71,36,11) embodiment, and decoding process includes the following step:

Digital signal is provided, utilizes digital signal to calculate a plurality of known features values, if during wrong generation, set wrong number and the error correcting capability value of occuring.Such as step S200.In this step, digital signal can calculate a plurality of known features values via equation 2, and wherein wrong number occurs is parameter v to hypothesis, and initial setting up numerical value is 1, and the error correcting capability value of quadratic residue code (71,36,11) is 5.

S _i=r (β ⁱ)=e (β ⁱ)=e ₀+ e ₁β ¹+ ... + e _N-1β ^N-1(equation 2)

S in the equation 2 wherein _iBe the known features value, β is GF (2 ³⁵) primary n root (Primitive nth Root of Unity) in the limited body.

Utilize a plurality of known features values to calculate a plurality of unknown characteristics values (step S210) of quadratic residue code (71,36,11).The unknown characteristics value S of quadratic residue code (71,36,11) ₇Account form is described as follows:

Case0：S ₇ ⁽⁰⁾＝0。

Case1：S ₇ ⁽¹⁾＝S ₁ ⁷。

Case2：I ₂＝{0，1，2}，J ₂＝{0，1，5}。

$S(I_2,J_2)=\left[\begin{array}{ccc}{S}_0& S_1& S_2\\ S_1& S_2& S_3\\ S_5& S_6& S_7^{\left(2\right)}\end{array}\right].$

S ₇ ⁽²⁾＝(S ₁S ₆+S ₂S ₅)+(S ₃S ₅/S ₁)。

Case3：I ₃＝{0，1，2，3}，J ₃＝{0，1，2，4}。

$S(I_3,J_3)=\left[\begin{array}{cccc}{S}_0& S_1& S_2& S_3\\ S_1& S_2& S_3& S_4\\ S_2& S_3& S_4& S_5\\ S_4& S_5& S_7^{\left(3\right)}& S_8\end{array}\right].$

Case4：I ₄＝{0，1，2，4，7}，J ₄＝{0，1，2，8，36}。

Case5：I ₅ ¹＝{0，1，2，4，7，24}，J ₅ ¹＝{0，1，3，5，8，36}

I ₅ ²＝{3，4，5，12，15，45}，J ₅ ²＝{0，3，4，15，33，45}。

Be in five the situation in number of errors, can't be only gather with one group (I, J) and just can define the unknown characteristics value S that all number of errors are five situation ₇ ⁽⁵⁾These two groups of S (I ₅ ¹, J ₅ ¹), S (I ₅ ², J ₅ ²) its determinantal expansion of matrix is zero, obtains following two S ₇ ⁽⁵⁾Cubic equation f ₁(S ₇ ⁽⁵⁾)=0 and g ₁(S ₇ ⁽⁵⁾)=0, coefficient is comprised of the known features value.Make again F ₁(S ₇ ⁽⁵⁾)=gcd (f ₁(S ₇ ⁽⁵⁾), g ₁(S ₇ ⁽⁵⁾)), for the situation F of all five mistakes ₁(S ₇ ⁽⁵⁾) be an order polynomial all, make F ₁(S ₇ ⁽⁵⁾)=0 can unique unknown characteristics value S that defines ₇ ⁽⁵⁾

Utilize a plurality of known features values and a plurality of unknown characteristics value as known, via without anti-Gen Bailigen (Inverse-free Berlekamp Massey) algorithm mistake in computation position multinomial, such as step S220.Main in this step to utilize a plurality of known features values and a plurality of not known features value that obtain be known input value, further calculates error location polynomial with error location polynomial that can efficient acquisition cyclic code without anti-Gen Bailigen algorithm.Wherein, of the step S120 among decoding quadratic residue code (89,45, the 17) embodiment without anti-Gen Bailigen algorithm steps.

Whether misjudgment position the highest polynomial power is consistent with mistake generation number, such as step S230.In this step, to utilize to judge whether formula deg (σ (x))=v misjudgment position the highest polynomial power is identical with wrong generation number, wherein, deg (σ (x)) is the highest power of error location polynomial, and v such as aforementioned be the wrong number that occurs.

If misjudgment position the highest polynomial power is identical with the wrong numerical value that number occurs, the Xie Genyu that then calculates error location polynomial separates radical, such as step S240.In this step, the Xie Genyu of error location polynomial separates radical, and both utilize error polynomial to calculate in conjunction with the chien search algorithm.Chien search algorithm described herein, it is used for seeking limited body root of polynomial, and with other algorithm in comparison, the search that applies to the error location polynomial root has better result of calculation.

If number increase certain value when not identical, occurs with mistake in misjudgment position the highest polynomial power and the wrong numerical value that number occurs and whether decision error generation number exceeds the error correcting capability value, such as step S250.This step is that the numerical value with definite value is made as 1, and the error correcting capability value of quadratic residue code (71,36,11) be as discussed previously be 5.When mistake generation number exceeds the error correcting capability value, namely can't revise again the digital signal operation, then finish the algorithm program.Otherwise, if when whether wrong generation number did not exceed the error correcting capability value, then a plurality of known features values of Returning utilization recalculated a plurality of unknown characteristics values of quadratic residue code (71,36,11), such as step S210.

Whether the polynomial solution radical in misjudgment position is consistent with mistake generation number, such as step S260.First error location polynomial is separated the radical computing, judged that again the solution radical that calculates occurs with wrong whether number is consistent.The employed judgement formula of this step is u=v, and wherein u is the solution radical, and v is the aforesaid wrong number that occurs.

When judging that two numerical value as consistent, seek the polynomial wrong occurrence positions in position that makes mistake, and numerical value corresponding to the occurrence positions that corrects mistakes, such as step S270.In this step, utilize error location polynomial to seek out the digital signal mistake occurrence positions of input, and the numerical value correction that obtains the digital signal errors present is obtained correct value.

Otherwise, if judge two numerical value not when consistent, then return and mistake occured number increases certain value and whether decision error generation number exceeds the error correcting capability value, such as step S250.

Please also refer to Fig. 3, it is the decoding algorithm of quadratic residue code of the present invention, decoding quadratic residue code (79,40,15) embodiment, and analysis process includes the following step:

Digital signal is provided, utilizes digital signal to calculate a plurality of known features values, if during wrong generation, set wrong number and the error correcting capability value of occuring, such as step S300.In this step, digital signal can calculate a plurality of known features values via equation 3, and wherein wrong number occurs is parameter v to hypothesis, and initial setting up numerical value is 1, and the error correcting capability value of quadratic residue code (79,40,15) is 7.

S _i=r (β ⁱ)=e (β ⁱ)=e ₀+ e ₁β ¹+ ... + e _N-1β ^N-1(equation 3)

S in the equation 3 wherein _iBe the known features value, β is GF (2 ³⁹) primary n root (Primitive nth Root of Unity) in the limited body.

Utilize a plurality of known features values to calculate a plurality of unknown characteristics values of quadratic residue code (79,40,15), such as step S310.The unknown characteristics value S of quadratic residue code (79,40,15) ₃Account form is described as follows:

Case0：S ₃ ⁽⁰⁾＝0。

Case1：S ₃ ⁽¹⁾＝S ₁ ³。

Case2：I ₂＝{0，1，2}，J ₂＝{0，1，8}。

$S(I_2,J_2)=\left[\begin{array}{ccc}{S}_0& S_1& S_2\\ S_1& S_2& S_3^{\left(2\right)}\\ S_8& S_9& S_{10}\end{array}\right].$

S ₃ ⁽²⁾＝S ₁S ₂+(S ₁S ₁₀+S ₂S ₉/S ₈)。

Case3：I ₃＝{0，1，2，3}，J ₃＝{0，8，18，19}。

$S(I_3,J_3)=\left[\begin{array}{cccc}{S}_0& S_1& S_2& S_3^{\left(3\right)}\\ S_8& S_9& S_{10}& S_{11}\\ S_{18}& S_{19}& S_{20}& S_{21}\\ S_{19}& S_{20}& S_{21}& S_{22}\end{array}\right].$

Case4：I ₄＝{0，1，2，3，13}，J ₄＝{0，8，18，19，49}。

Case5：I ₅ ¹＝{2，3，4，19，25，73}，J ₅ ¹＝{0，6，7，17，19，48}

Case6：I ₆ ¹＝{1，2，3，4，5，64，72}，J ₆ ¹＝{0，1，8，17，18，19，20}

I ₆ ²＝{0，1，2，3，16，22，42}，J ₆ ²＝{0，2，3，9，10，20，22}。

Be in six the situation in number of errors, can't be only gather with one group (I, J) that just to define all number of errors be six unknown characteristics value S ₃ ⁽⁶⁾The determinantal expansion of this two groups (I, J) set all is nonlinear S ₃ ⁽⁶⁾Multinomial learns that by theorem 2 among decoding quadratic residue code (89,45, the 17) embodiment these determinants equal zero.In other words, these two multinomials have identical liner factor at least.Therefore, use Ou Ji Reed algorithm (Euclid ' s Algorithm) to obtain this two polynomial highest common factors.Find that through the simulation test of computer these two groups is that six situation can obtain liner factor and namely can define unique S for all number of errors ₃ ⁽⁶⁾Value.

Case7：I ₇ ¹＝{0，1，2，3，4，5，64，72}，J ₇ ¹＝{0，1，8，16，17，18，19，20}，I ₇ ²＝{0，1，2，3，4，16，18，24}，J ₇ ²＝{0，1，2，8，16，18，20，22}。

Be that six situation is identical with number of errors, S (I ₇ ¹, J ₇ ¹) and S (I ₇ ², J ₇ ²) determinantal expansion after we obtain two S that are respectively 83 powers and 153 powers ₃ ⁽⁷⁾Polynomial f ₂(S ₃ ⁽⁷⁾) and g ₂(S ₃ ⁽⁷⁾).Re-use Ou Ji Reed algorithm (Euclid ' s Algorithm) and obtain this two polynomial highest common factor F ₂(S ₃ ⁽⁷⁾)=gcd (f ₂(S ₃ ⁽⁷⁾), g ₂(S ₃ ⁽⁷⁾)).All code words of receiving via Computer Simulation Test, F ₂(S ₃ ⁽⁷⁾) all be an order polynomial.Learn f by theorem 1 among decoding quadratic residue code (89,45, the 17) embodiment ₂(S ₃ ⁽⁷⁾)=0 and g ₂(S ₃ ⁽⁷⁾So)=0 is F ₂(S ₃ ⁽⁷⁾) also be 0.Therefore number of errors is seven original unknown characteristics value S ₃ ⁽⁷⁾Can be by unique definition.

Utilize a plurality of known features values and a plurality of unknown characteristics value as known, via without anti-Gen Bailigen (Inverse-free Berlekamp Massey) algorithm mistake in computation position multinomial, such as step S320.Main in this step to utilize a plurality of known features values and a plurality of not known features value that obtain be known input value, further calculates error location polynomial with error location polynomial that can efficient acquisition cyclic code without anti-Gen Bailigen algorithm.Wherein, of the step S120 among decoding quadratic residue code (89,45, the 17) embodiment without anti-Gen Bailigen algorithm steps.

Whether misjudgment position the highest polynomial power is consistent with mistake generation number, such as step S330.In this step, to utilize to judge whether formula deg (σ (x))=v misjudgment position the highest polynomial power is identical with wrong generation number, wherein, deg (σ (x)) is the highest power of error location polynomial, and v such as aforementioned be the wrong number that occurs.

If misjudgment position the highest polynomial power is identical with the wrong numerical value that number occurs, the Xie Genyu that then calculates error location polynomial separates radical, such as step S340.In this step, the Xie Genyu of error location polynomial separates radical, and both utilize error polynomial to calculate in conjunction with the chien search algorithm.Chien search algorithm described herein, it is used for seeking limited body root of polynomial, and with other algorithm in comparison, the search that applies to the error location polynomial root has better result of calculation.

If number increase certain value when not identical, occurs with mistake in misjudgment position the highest polynomial power and the wrong numerical value that number occurs and whether decision error generation number exceeds the error correcting capability value, such as step S350.This step be that the numerical value with definite value is made as 1, and the error correcting capability value of quadratic residue code (79,40,15) is 7.When mistake generation number exceeds the error correcting capability value, namely can't revise again the digital signal operation, then finish the algorithm program.Otherwise, if when whether wrong generation number did not exceed the error correcting capability value, then a plurality of known features values of Returning utilization calculated a plurality of unknown characteristics values of quadratic residue code (79,40,15), such as step S310.

Whether the polynomial solution radical in misjudgment position is consistent with mistake generation number, such as step S360.First error location polynomial is separated the radical computing, judged that again the solution radical that calculates occurs with wrong whether number is consistent.The employed judgement formula of this step is u=v, and wherein u is the solution radical, and v is the aforesaid wrong number that occurs.

When judging that two numerical value as consistent, seek the polynomial wrong occurrence positions in position that makes mistake, and numerical value corresponding to the occurrence positions that corrects mistakes, such as step S370.In this step, utilize error location polynomial to seek out the digital signal mistake occurrence positions of input, and the numerical value correction that obtains the digital signal errors present is obtained correct value.

Otherwise, if judge two numerical value not when consistent, then return and mistake occured number increases certain value and whether decision error generation number exceeds the error correcting capability value, such as step S350.

Although the present invention with aforesaid preferred embodiment openly as above; so it is not to limit the present invention, any those of ordinary skills, without departing from the spirit and scope of the present invention; the equivalence of doing to change with retouching is replaced, all in scope of patent protection of the present invention.

Claims (6)

1. the decoding algorithm of a quadratic residue code is applicable to have a computer installation of a decoder, and the decoding algorithm of this quadratic residue code comprises the following step:

One digital signal is provided, utilizes this digital signal to calculate a plurality of known features values, if wrong number and the error correcting capability value of occuring set in wrong generation;

Utilize described a plurality of known features value to calculate a plurality of unknown characteristics values;

Utilize described a plurality of known features value and described a plurality of unknown characteristics value as known, calculate an error location polynomial;

Whether the highest power of judging this error location polynomial is consistent with this mistake generation number;

Result of determination is yes, then calculates at least one solution root and at least one solution radical of this error location polynomial;

Result of determination is no, should increase certain value and judge whether this mistake generation number exceeds this error correcting capability value by mistake generation number, if judgement does not exceed this error correcting capability value, then returns this and utilizes described a plurality of known features value to calculate a plurality of unknown characteristics value steps;

Whether this solution radical of judging this error location polynomial is consistent with this mistake generation number;

Result of determination is yes, then seeks out at least one wrong occurrence positions of this error location polynomial, and revises numerical value corresponding to this mistake occurrence positions; And

Be judged to be when no, returning this mistake generation number increases certain value and judges whether this mistake generation number exceeds this error correcting capability value step.

2. the decoding algorithm of quadratic residue code as claimed in claim 1, wherein the decoding algorithm of this quadratic residue code is applicable to all quadratic residue codes.

3. the decoding algorithm of quadratic residue code as claimed in claim 1, wherein the decoding algorithm of this quadratic residue code is applicable to quadratic residue code (89,45,17).

4. the decoding algorithm of quadratic residue code as claimed in claim 1, initial setting up numerical value that wherein should mistake generation number is 1, and this definite value is 1.

5. the decoding algorithm of quadratic residue code as claimed in claim 1, wherein utilize described a plurality of known features value and described a plurality of unknown characteristics value as known, calculate in this error location polynomial step, this error location polynomial is take described a plurality of known features values and described a plurality of unknown characteristics value as known input value, calculates and gets without anti-Gen Bailigen algorithm with one.

6. the decoding algorithm of quadratic residue code as claimed in claim 1, wherein at least one solution root of this error location polynomial and at least one solution radical are take this error location polynomial as the basis, calculate it with a chien search algorithm.

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