CN104915699A - Matrix two-dimensional code RS decoding error correction method based on improved GS algorithm - Google Patents

Abstract

The invention provides a matrix two-dimensional code RS decoding error correction method based on an improved GS algorithm. The improved GS algorithm can reduce the complexity of interpolation, so that the efficiency of the algorithm can be improved. Meanwhile, the decoding error correction method studies two encoding modes and a transformation relation thereof of RS codes, so that correction can be carried out on the RS codes by using the transformation process of the two encoding modes, and two-dimensional codes are also enabled to restore original information correctly when circumstances such as dropping, stains, punching, local damages and the like occur.

Description

Based on the matrix two-dimensional code RS decoding error correction method of the GS algorithm improved

Technical field

The present invention relates to Quick Response Code field, more specifically, relating to a kind of matrix two-dimensional code RS decoding error correction method of the GS algorithm based on improving.

Background technology

Quick Response Code has that memory data output is large, confidentiality is high, traceability is high, damage resistance is strong, redundant is large, cost is cheap, interactive strong, the feature such as experience property is good, and can organically combine with the mobile terminal such as smart mobile phone better.According to the coding principle of Quick Response Code, the difference of planform, Quick Response Code can be divided into capable row's formula Quick Response Code and matrix two-dimensional code.Wherein matrix two-dimensional code is higher than the information density of row row formula Quick Response Code, and therefore its range of application is comparatively extensive, and representational matrix two-dimensional code has DataMatrix, MaxiCode, CodeOne, QRCode etc.The introducing of error correcting technique is one of principal feature of Quick Response Code, and owing to using RS systematic code in conventional matrix two-dimensional code, therefore involved decoding error correction algorithm is RS decoding error correction algorithm.

Summary of the invention

The invention provides a kind of matrix two-dimensional code RS decoding error correction method of the GS algorithm based on improving, the GS algorithm improved can reduce the complexity of interpolation, therefore the efficiency of algorithm can be improved, the error correction method of decoding is simultaneously studied two of RS code kinds of coded systems and transforming relationship thereof, thus utilize the conversion process of two kinds of coded systems to correct RS code, make Quick Response Code occurring coming off, stain, the situation such as perforation and local damage time, correctly can reduce raw information.

For realizing above goal of the invention, the technical scheme of employing is:

Based on a matrix two-dimensional code RS decoding error correction method for the GS algorithm improved, the GS algorithm wherein improved comprises Kotter interpolation algorithm and the Roth-Ruckenstein factoring algorithm of improvement, it is characterized in that: described error correction method comprises the following steps:

S1. mask process is gone to image in 2 D code, obtain RS (n, the k) code after removing mask, fill (255-n) individual zero in the last position of RS (n, k) code, structure RS (255,255+k-n) code;

S2. code word RS (255,255+k-n) is passed through in conjunction with finite field gf (q ^m) middle nonzero element (x ₀, x ₁..., x _255+k-n-1) form 255+k-n interpolation point (x ₀, r ₀), (x ₁, r ₁) ..., (x _255+k-n-1, r _255+k-n-1), then based on (1, k-1)-weighting dictionary inverted sequence table by each interpolation point at least interpolation build a binary polynomial for m time, wherein m is interpolation severe;

S3. utilize the Kotter interpolation algorithm of improvement to ask for minimal polynomial, detailed process is as follows:

S31. first formula is passed through initialization one group of binary polynomial, wherein for initialized jth bar binary polynomial, l _mfor the number of this group binary polynomial, for the set of the binary polynomial composition after initialization, i _kfor iterations, now i _k=0;

S32. formula is passed through cancellation set interior first rank are greater than the binary polynomial of C, wherein $C=n\left(\begin{array}{c}m+1\\ 2\end{array}\right);$

S33. formula is passed through right in the Hasse mixed partial derivative of each binary polynomial calculate, then judge set in the Hasse mixed partial derivative of all binary polynomials whether all equal 0, if all equal 0, then carry out step S36, if be not congruent to 0, carry out step S34;

S34. ask for the minimal polynomial in this group binary polynomial, be shown below:

$f=\underset{j\∈J_{i_k}}{\min }{g}_{i_k,j}$

$j^{*}=\arg \underset{j\∈J_{i_k}}{\min }{g}_{i_k,j}$

Wherein f is the minimal polynomial in this group binary polynomial, j ^*for the sequence number that minimal polynomial is corresponding;

S35. carry out conversion amendment to the minimal polynomial in this group binary polynomial, formula is as follows:

$g_{i_k+1,j^{*}}={[\mathrm{xf},f]}_{D_{i_k}}={\mathrm{\Δ}}_{j^{*}}(x-x_i)f,j=j^{*},$

Remaining binary polynomial also carries out conversion amendment, and formula is as follows:

$g_{i_k+1,j}={[g_{i_k,j},f]}_{D_{i_k}}={\mathrm{\Δ}}_{j^{*}}{g}_{i_k,j}-{\mathrm{\Δ}}_{j}f,j\≠j^{*};$

S36. the process choosing another group binary polynomial repetition step S31 ~ S35 carries out asking for of this group minimal polynomial, and makes i _k=i _k+ 1;

If S37. i _k=C then stops iteration, now respectively organizes the minimal polynomial g of binary polynomial _c,jcomposition set G _c, by formula Q (x, y)=min{g _c,j| g _c,j∈ G _cminimal polynomial Q (x, y) in C group binary polynomial is solved;

S4. after trying to achieve Q (x, y), Roth-Ruckenstein factoring algorithm is utilized to carry out decomposing message polynomial m'(x corresponding to acquisition RS code Frequency Domain Coding to Q (x, y));

S5. to message polynomial m'(x) adopt Frequency Domain Coding mode to carry out n position coding, get coding codeword n-th-k+1 after obtaining coding codeword and generate message polynomial m (x) corresponding to time domain coding to n bit word, time domain coding mode is adopted to encode to m (x), RS (n, the k) code of correction can be obtained.

Preferably, in step S4, obtain the message polynomial m'(x that RS code Frequency Domain Coding is corresponding) process specific as follows:

After using Roth-Ruckenstein factoring algorithm to decompose Q (x, y), obtain the factor of some shapes as y-p (x), wherein p (x) is polynomial of one indeterminate, decomposes the polynomial of one indeterminate obtained and forms set L:

L={p (x): (y-p (x)) | Q (x, y) and deg p (x) < k}

Deg p (x) < k represents that the number of times of p (x) is less than k, Frequency Domain Coding mode is adopted to encode to polynomial of one indeterminate all in set L, the code word and RS (n that obtain encoding, k) code compares, and polynomial of one indeterminate corresponding to the minimum code word of Hamming distance is message polynomial m'(x).

Preferably, in step S2, the process of the binary polynomial J (x, y) of structure can be represented by the formula:

$\begin{array}{c}J(x,y)=\underset{\mathrm{lod}\left(J(x,y)\right)}{\min }\{J(x,y)\&Element;F_q[x,y\left]\right|D_{\mathrm{\&alpha;\&beta;}}J(x_i,r_i)=0,i=\mathrm{0,1},...,n-1,\\ \mathrm{\&alpha;}+\mathrm{\&beta;}<m(\mathrm{\&alpha;},\mathrm{\&beta;}\&Element;N)\}\end{array}$

D _{α β}j (x _i, r _i) represent that binary polynomial J (x, y) is at interpolation point (x _i, y _i) (α, β) rank Hasse mixed partial derivative, F _q[x, y] for independent variable be the Polynomial Ring with Two of x and y, F _q[x, y] represents that coefficient belongs to finite field gf (q ^m) in element, F _q[x, y] can be expressed as with formula wherein f _ab∈ GF (q ^m).

Preferably, in step S5, to message polynomial m'(x) to adopt Frequency Domain Coding mode to carry out the process of n position coding specific as follows:

$(c_0,c_1,...,c_{n-1})\stackrel{\mathrm{\&Delta;}}{=}f\left(m^{\&prime;}\left(x\right)\right)=(m\left(1\right),m\left(\mathrm{\&alpha;}\right),m\left({\mathrm{\&alpha;}}^2\right),...,m\left({\mathrm{\&alpha;}}^{n-1}\right))$

Wherein m'(x)=m ₀+ m ₁x+...+m _k-1x ^k-1, m'(x) and ∈ GF (q ^m);

(c ₀, c ₁..., c _n-1) be coding codeword, α is finite field gf (q ^m) primitive element.

Preferably, in step S5, the process of generator polynomial g (x) can be expressed as follows:

g(x)＝(x-α ^l)(x-α ^l+1)...(x-α ^l+2t-1)

Wherein n-k=2t, l are integer, and α is primitive element, and the generator matrix that m (x) is corresponding can be expressed as:

$G_2=\left[\begin{array}{ccccc}1& 0& ...& 0& x^{n-1}\left(\mathrm{mod}g\left(x\right)\right)\\ 0& 1& ...& 0& x^{n-2}\left(\mathrm{mod}g\left(x\right)\right)\\ .& .& & & .\\ .& .& ...& 0& .\\ .& .& & & .\\ 0& 0& ...& 0& x^{n-k}\left(\mathrm{mod}g\left(x\right)\right)\end{array}\right]$

Mod g (x) represents generator polynomial g (x) remainder, obtains result and is error correction codeword polynome;

Codeword polynome c (x) after the above-mentioned coding can obtaining message polynomial m (x) correspondence of through type,

c(x)＝m(x)+[m(x)x ^n-kmodg(x)]

The coefficient of codeword polynome c (x) is the codeword vector of time domain coding, by codeword vector, can obtain RS (n, the k) code of correction.

Compared with prior art, the invention has the beneficial effects as follows:

The invention provides a kind of matrix two-dimensional code RS decoding error correction method of the GS algorithm based on improving, the GS algorithm improved can reduce the complexity of interpolation, therefore the efficiency of algorithm can be improved, the error correction method of decoding is simultaneously studied two of RS code kinds of coded systems and transforming relationship thereof, therefore the conversion process of two kinds of coded systems can be utilized to correct RS code, make Quick Response Code occurring coming off, stain, the situation such as perforation and local damage time, also correctly can reduce raw information.

Accompanying drawing explanation

Fig. 1 is the conventional conversion method process flow diagram of RS code two kinds of coded systems.

Fig. 2 is that GS algorithm and list decoding are to the decoding process flow diagram of time domain coding gained code word.

Fig. 3 is the process flow diagram of decoding error correction method.

Embodiment

Accompanying drawing, only for exemplary illustration, can not be interpreted as the restriction to this patent;

Below in conjunction with drawings and Examples, the present invention is further elaborated.

Embodiment 1

Before being described technical scheme of the present invention, first carry out brief introduction to RS code, RS code has two kinds of coded systems: Frequency Domain Coding mode and time domain coding mode.These two kinds of coded systems are by assuming the forms of time and space of code word by original codeword vector c, regard the codeword vector C of Galois field Fourier transform (GFFT) gained corresponding for c as frequency domain form, can prove that both are of equal value according to the character of GFFT conversion.Therefore these two kinds of coded systems can be seen as time domain coding (it doesn't matter even if with the time) and Frequency Domain Coding respectively.The corresponding time domain coding of RS systematic code in Quick Response Code, and the corresponding Frequency Domain Coding of the GS algorithm improved, the RS code word simultaneously removed after mask due to two-dimensional code symbol is shortened code, first will study the transforming relationship between two kinds of coded systems when this makes the code word after removing mask with the GS algorithm improved to two-dimensional code symbol carry out error-correcting decoding.

In prior art, when carrying out the conversion between two kinds of coded systems, as shown in Figure 1, the method comprised the following steps usually is used:

Step 1: by the code word c of time domain coding ₂namely c' is become before lower position ₂;

Step 2: the generator matrix G obtaining Frequency Domain Coding respectively ₁with the generator matrix G of time domain coding ₂;

Step 3: to G ₂first spin upside down left and right reversion again and obtain G ₂';

Step 4: utilization linear algebra asks the method for solution of equations to try to achieve matrix A (n × n) and B (n × n) makes G' ₂a=G ₁or G ₁b=G' ₂, i.e. c' ₂a=c ₁or c ₁b=c' ₂, thus realize the conversion between two kinds of coding codewords.

But when the code word received is wrong, the method of above-mentioned conversion is obviously unworkable, therefore need to find another method for transformation, if list decoding is to the interpretation method of time domain coding gained code word, this method point shortened code and non-truncated code two kinds of situations are discussed, because GS algorithm is the one of list decoding, and codeword coding principle is identical, therefore the method to GS algorithm and list decoding all applicable, as shown in Figure 2, said method comprising the steps of.

Step 1: judge whether RS (n, k) code is shortened code;

Step 2: if not shortened code, then turn to step 3; If so, then q is filled to the information bit of shortened code ^m-1-n zero, structure RS (q ^m-1, k+q ^m-1-n) code;

Step 3: decoding is carried out to gained RS code word GS algorithm and list decoding and obtains message polynomial m'(x);

Step 4: to message polynomial m'(x) carry out n position coding with RS Frequency Domain Coding, get the n-th-k+1 to n position coding codeword and namely obtain required message polynomial m (x) (message polynomial that time domain coding is corresponding).

Above-mentioned step of converting occurs coming off at Quick Response Code, stain, the situation such as perforation and local damage time, also can normally transform, on the basis of above method, the invention provides a kind of matrix two-dimensional code RS decoding error correction method of the GS algorithm based on improving, the GS algorithm wherein improved comprises Kotter interpolation algorithm and the Roth-Ruckenstein factoring algorithm of improvement, as shown in Figure 3, error correction method comprises the following steps:

S1. mask process is gone to image in 2 D code, obtain the RS (n after removing mask, k) code, due to RS (n, k) code is shortened code, therefore needs to fill (255-n) individual zero in the last position of RS (n, k) code, structure RS (255,255+k-n) code;

S2. code word RS (255,255+k-n) is passed through in conjunction with finite field gf (q ^m) middle nonzero element (x ₀, x ₁..., x _255+k-n-1) form 255+k-n interpolation point (x ₀, r ₀), (x ₁, r ₁) ..., (x _255+k-n-1, r _255+k-n-1), then based on (1, k-1)-weighting dictionary inverted sequence table by each interpolation point at least interpolation build a binary polynomial for m time, wherein m is interpolation severe;

The process of the binary polynomial J (x, y) built can be represented by the formula:

$\begin{array}{c}J(x,y)=\underset{\mathrm{lod}\left(J(x,y)\right)}{\min }\{J(x,y)\&Element;F_q[x,y\left]\right|D_{\mathrm{\&alpha;\&beta;}}J(x_i,r_i)=0,i=\mathrm{0,1},...,n-1,\\ \mathrm{\&alpha;}+\mathrm{\&beta;}<m(\mathrm{\&alpha;},\mathrm{\&beta;}\&Element;N)\}\end{array}$

D _{α β}j (x _i, r _i) represent that binary polynomial J (x, y) is at interpolation point (x _i, y _i) (α, β) rank Hasse mixed partial derivative, F _q[x, y] for independent variable be the Polynomial Ring with Two of x and y, F _q[x, y] represents that coefficient belongs to finite field gf (q ^m) in element, F _q[x, y] can be expressed as with formula wherein f _ab∈ GF (q ^m).

S3. utilize the Kotter interpolation algorithm of improvement to ask for minimal polynomial, detailed process is as follows:

S31. first formula is passed through initialization one group of binary polynomial, wherein for initialized jth bar binary polynomial, l _mfor the number of this group binary polynomial, for the set of the binary polynomial composition after initialization, i _kfor iterations, now i _k=0;

S32. formula is passed through cancellation set interior first rank are greater than the binary polynomial of C, wherein $C=n\left(\begin{array}{c}m+1\\ 2\end{array}\right);$

S33. formula is passed through right in the Hasse mixed partial derivative of each binary polynomial calculate, then judge set in the Hasse mixed partial derivative of all binary polynomials whether all equal 0, if all equal 0, then carry out step S36, if be not congruent to 0, carry out step S34;

S34. ask for the minimal polynomial in this group binary polynomial, be shown below:

$f=\underset{j\&Element;J_{i_k}}{\min }{g}_{i_k,j}$

$j^{*}=\arg \underset{j\&Element;J_{i_k}}{\min }{g}_{i_k,j}$

Wherein f is the minimal polynomial in this group binary polynomial, j ^*for the sequence number that minimal polynomial is corresponding;

S35. carry out conversion amendment to the minimal polynomial in this group binary polynomial, formula is as follows:

$g_{i_k+1,j^{*}}={[\mathrm{xf},f]}_{D_{i_k}}={\mathrm{\&Delta;}}_{j^{*}}(x-x_i)f,j=j^{*},$

Remaining binary polynomial also carries out conversion amendment, and formula is as follows:

$g_{i_k+1,j}={[g_{i_k,j},f]}_{D_{i_k}}={\mathrm{\&Delta;}}_{j^{*}}{g}_{i_k,j}-{\mathrm{\&Delta;}}_{j}f,j\&NotEqual;j^{*};$

S36. the process choosing another group binary polynomial repetition step S31 ~ S35 carries out asking for of this group minimal polynomial, and makes i _k=i _k+ 1;

If S37. i _k=C then stops iteration, now respectively organizes the minimal polynomial g of binary polynomial _c,jcomposition set G _c, by formula Q (x, y)=min{g _c,j| g _c,j∈ G _cminimal polynomial Q (x, y) in C group binary polynomial is solved;

S4., after trying to achieve Q (x, y), Roth-Ruckenstein factoring algorithm is utilized to carry out decomposing message polynomial m'(x corresponding to acquisition RS code Frequency Domain Coding to Q (x, y)), specific as follows:

After using Roth-Ruckenstein factoring algorithm to decompose Q (x, y), obtain the factor of some shapes as y-p (x), wherein p (x) is polynomial of one indeterminate, decomposes the polynomial of one indeterminate obtained and forms set L:

L={p (x): (y-p (x)) | Q (x, y) and deg p (x) < k}

Deg p (x) < k represents that the number of times of p (x) is less than k, Frequency Domain Coding mode is adopted to encode to polynomial of one indeterminate all in set L, the code word and RS (n that obtain encoding, k) code compares, and polynomial of one indeterminate corresponding to the minimum code word of Hamming distance is message polynomial m'(x).

The false code of Roth-Ruchenstein algorithm can simply be described as:

Input: interpolation polynomial Q (x, y), D=k-1 (maximum times of p (x))

Export: a series of degree of polynomial is less than or equal to the set L of polynomial of one indeterminate p (x) of D, and each polynomial expression meets (y-p (x)) | Q (x, y)

BEGIN

π[0]＝0；deg[0]＝-1；Q ₀(x,y)＝＜＜Q(x,y)＞＞；t＝1；s＝0；

rothrucktree(s)；

END

Each coefficient of the recursive resolve polynomial of one indeterminate of rothrucktree (s) // from s

BEGIN

IF Q _s(x, 0)=0, then export p _[s](x);

ELSEIF(deg[s]＜D)

R＝Rootlist[Q _s(0,y)]

FOR(α∈R)

v＝t；t＝t+1；

π[v]＝s；deg[v]＝deg[s]+1；coff[v]＝α；

Q _v(x,y)＝＜＜Q _s(x,xy+α)＞＞；

rothrucktree(v)；

END

Wherein π [s] represents the root of node s, and deg [s] represents the number of times of node s, and coff [s] represents the multinomial coefficient at node s place, p _[s](x)=coff [s] x ^{deg [s]}+ coff [π [s]] x ^{deg [π [s]]}+ ....

S5. to message polynomial m'(x) adopt Frequency Domain Coding mode to carry out n position coding, get coding codeword n-th-k+1 after obtaining coding codeword and generate message polynomial m (x) corresponding to time domain coding to n bit word, time domain coding mode is adopted to encode to m (x), RS (n, the k) code of correction can be obtained.

Wherein to message polynomial m'(x) to adopt Frequency Domain Coding mode to carry out the process of n position coding specific as follows:

$(c_0,c_1,...,c_{n-1})\stackrel{\mathrm{\&Delta;}}{=}f\left(m^{\&prime;}\left(x\right)\right)=(m\left(1\right),m\left(\mathrm{\&alpha;}\right),m\left({\mathrm{\&alpha;}}^2\right),...,m\left({\mathrm{\&alpha;}}^{n-1}\right))$

Wherein m'(x)=m ₀+ m ₁x+...+m _k-1x ^k-1, m'(x) and ∈ GF (q ^m);

(c ₀, c ₁..., c _n-1) be coding codeword, α is finite field gf (q ^m) primitive element and code word size n=q ^m-1.

Meanwhile, the process of generator polynomial g (x) can be expressed as follows:

g(x)＝(x-α ^l)(x-α ^l+1)...(x-α ^l+2t-1)

Wherein n-k=2t, l are integer, and α is primitive element, and the generator matrix that m (x) is corresponding can be expressed as:

$G_2=\left[\begin{array}{ccccc}1& 0& ...& 0& x^{n-1}\left(\mathrm{mod}g\left(x\right)\right)\\ 0& 1& ...& 0& x^{n-2}\left(\mathrm{mod}g\left(x\right)\right)\\ .& .& & & .\\ .& .& ...& 0& .\\ .& .& & & .\\ 0& 0& ...& 0& x^{n-k}\left(\mathrm{mod}g\left(x\right)\right)\end{array}\right]$

Mod g (x) represents generator polynomial g (x) remainder, obtains result and is error correction codeword polynome;

Codeword polynome c (x) after the above-mentioned coding can obtaining message polynomial m (x) correspondence of through type,

c(x)＝m(x)+[m(x)x ^n-kmodg(x)]

The coefficient of codeword polynome c (x) is the codeword vector of time domain coding, by codeword vector, can obtain RS (n, the k) code of correction.

In such scheme, for arbitrary RS (n, k) code, the maximum error correcting capability of decoding error correction method is:

$t_{\max }=n-\sqrt{n(k-1)}-1$

For QR code version 1-H symbol, concerning this symbol, finite field gf (2 ⁸) on (26,9,8) RS code be used for error correction.After conventional letter removes mask, then the generator polynomial that this code word is corresponding is:

g(x)＝(x-1)(x-α)(x-α ²)...(x-α ¹⁶)＝x ¹⁷+α ⁴³x ¹⁶+α ¹³⁹x ¹⁵+α ²⁰⁵x ¹⁴+α ⁷⁸x ¹³+α ⁴³x ¹²+α ²³⁹x ¹¹+α ¹²³x ¹⁰+α ²⁰⁶x ⁹+α ²¹⁴x ⁸+α ¹⁴⁷x ⁷+α ²⁴x ⁶+α ⁹⁹x ⁵+α ¹⁵⁰x ⁴+α ³⁹x ³+α ²⁴³x ²+α ¹⁶³x ²+α ¹³⁶

The syndrome decoding algorithm used in current Quick Response Code can only correct 8 mistakes at the most to (26,9,8) RS code, but the decoding error correction method that the present invention proposes can break through this restriction, as shown in table 1 according to different its error correcting capabilities of interpolation severe m:

T in table 1RS (26,9,8) code _malong with the change of interpolation severe m

Interpolation severe m	1	2	6	30
					Error correcting capability t m	9	10	11	12

As can be seen from Table 1, with the GS algorithm improved to RS (26,9,8) code carries out error-correcting decoding, error correction code word number increases gradually along with interpolation severe m, and its error correcting capability is just greater than the error correcting capability of syndrome decoding algorithm as m=1, this makes Quick Response Code can reduce raw information more accurate, better when breakage.

In order to be described technical scheme of the present invention better, the present embodiment has carried out sufficient explanation by the correction procedure of example 1, example 2 pairs of decoding error correction methods.

Example 1

Example 1 illustrates how to carry out error-correcting decoding with decoding error correction method to the RS systematic code in Quick Response Code for interpolation severe 2.

Code word after conventional letter removes mask is:

R＝(r ₀,r ₁,...,r ₂₅)＝(α ¹⁹,α ³,α ²¹⁴,α ¹⁸,α ¹⁶⁹,α ³¹,α ¹⁹⁶,α ¹⁰⁶,α ²³⁸,α ⁷⁵,α ¹⁹²,α ³³,α ¹⁶¹,α ¹⁸⁵,α ⁹⁶,α ¹⁵³,α ⁴³,α ⁵¹,α ²⁷,α ²⁴⁸,α ²⁴,α ¹⁹⁵,α ¹²³,α ²²⁰,α ¹⁷,α ³²)

Namely can form 255 interpolation points altogether, be respectively:

(1, α ¹⁹), (α, α ³), (α ², α ²¹⁴), (α ³, α ¹⁸), (α ⁴, α ¹⁶⁹), (α ⁵, α ³¹) (α ⁶, α ¹⁹⁶), (α ⁷, α ¹⁰⁶), (α ⁸, α ²³⁸), (α ⁹, α ⁷⁵), (α ¹⁰, α ¹⁹²), (α ¹¹, α ³³), (α ¹², α ¹⁶¹), (α ¹³, α ¹⁸⁵), (α ¹⁴, α ⁹⁶), (α ¹⁵, α ¹⁵³), (α ¹⁶, α ⁴³), (α ¹⁷, α ⁵¹), (α ¹⁸, α ²⁷), (α ¹⁹, α ²⁴⁸), (α ²⁰, α ²⁴), (α ²¹, α ¹⁹⁵), (α ²², α ¹²³), (α ²³, α ²²⁰), (α ²⁴, α ¹⁷), (α ²⁵, α ³²) and (α ²⁶, 0), (α ²⁷, 0) ..., (α ²⁵⁴, 0);

255 interpolated points form RS (255 altogether, 238,8), the GS algorithm that use improves solves can obtain message polynomial m'(x to the polynomial expression built), use Frequency Domain Coding mode to message polynomial m'(x) carry out 26 codings, and high 9 that get gained code word generate message polynomial m (x) corresponding to time domain coding:

M (x)=α ⁵¹+ α ²²x+ α ²⁴³x ²+ α ²⁴⁷x ³+ α ¹⁹⁵x ⁴+ α ¹²³x ⁵+ α ²²⁰x ⁶+ α ¹⁷⁶x ⁷+ α ³²x ⁸m (x) time domain coded system is encoded, coding codeword c can be obtained, that is:

c＝(c ₀,c ₁,...,c ₂₅)＝(α ³⁹,α ³,α ²¹⁴,α ¹⁹⁸,α ¹⁶⁹,α ³¹,α ¹⁶⁹,α ¹⁰⁶,α ²²⁸,α ⁵⁷,α ¹⁹²,α ³³,α ¹⁶¹,α ¹⁸⁴,α ⁹⁶,α ¹⁵³,α ⁴³,α ⁵¹,α ²²,α ²⁴³,α ²⁴⁷,α ¹⁹⁵,α ¹²³,α ²²⁰,α ¹⁷⁶,α ³²)

Contrast known by coding codeword c and formula (12), correct for QR code version 1-H symbolic solution except 10 wrong code words after mask by the decoding error correction method of interpolation severe m=2, this can not accomplish with current syndrome decoding algorithm.

Example 2

Suppose that the shortened code of RS (7,4) is RS (5,2):

Suppose that the code word received is c' ₃=(α ⁵, α ⁴, α ⁴, α, α ²), then at c' ₃last position fill and make for two 0 it become 7 original bit word, i.e. c " ₃=(α ⁵, α ⁴, α ⁴, α, α ², 0,0), use the GS algorithm improved to c " ₃carry out decoding, obtain message polynomial m'(x), m'(x)=σ ³x+x ²+ σ ⁶x ³, adopt Frequency Domain Coding mode to encode to message polynomial:

$(0,{\mathrm{\&alpha;}}^3,1,{\mathrm{\&alpha;}}^6)\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& \mathrm{\&alpha;}& {\mathrm{\&alpha;}}^2& {\mathrm{\&alpha;}}^3& {\mathrm{\&alpha;}}^4\\ 1& {\mathrm{\&alpha;}}^2& {\mathrm{\&alpha;}}^4& {\mathrm{\&alpha;}}^6& {\mathrm{\&alpha;}}^3\\ 1& {\mathrm{\&alpha;}}^3& {\mathrm{\&alpha;}}^6& {\mathrm{\&alpha;}}^2& {\mathrm{\&alpha;}}^6\end{array}\right]=({\mathrm{\&alpha;}}^5,{\mathrm{\&alpha;}}^4,{\mathrm{\&alpha;}}^4,\mathrm{\&alpha;},{\mathrm{\&alpha;}}^6)$

Get (α ⁵, α ⁴, α ⁴, α, α ⁶) high two bit word generate message polynomial m (x) corresponding to time domain codings, m (x)=α+α ⁶x, to m (x) adopt time domain coding mode to carry out coding codeword (α that coding can obtain RS (5,2) ⁵, α ⁴, α ⁴, α, α ⁶), with c' ₃compare, known its correct for a mistake.

The invention provides a kind of matrix two-dimensional code RS decoding error correction method of the GS algorithm based on improving, the GS algorithm improved can reduce the complexity of interpolation, therefore the efficiency of algorithm can be improved, the error correction method of decoding is simultaneously studied two of RS code kinds of coded systems and transforming relationship thereof, therefore the conversion process of two kinds of coded systems can be utilized to correct RS code, make Quick Response Code occurring coming off, stain, the situation such as perforation and local damage time, also correctly can reduce raw information.

Obviously, the above embodiment of the present invention is only for example of the present invention is clearly described, and is not the restriction to embodiments of the present invention.For those of ordinary skill in the field, can also make other changes in different forms on the basis of the above description.Here exhaustive without the need to also giving all embodiments.All any amendments done within the spirit and principles in the present invention, equivalent to replace and improvement etc., within the protection domain that all should be included in the claims in the present invention.

Claims (5)

1. the matrix two-dimensional code RS decoding error correction method based on the GS algorithm improved, the GS algorithm wherein improved comprises Kotter interpolation algorithm and the Roth-Ruckenstein factoring algorithm of improvement, it is characterized in that: described error correction method comprises the following steps:

S1. mask process is gone to image in 2 D code, obtain RS (n, the k) code after removing mask, fill (255-n) individual zero in the last position of RS (n, k) code, structure RS (255,255+k-n) code;

S2. code word RS (255,255+k-n) is passed through in conjunction with finite field gf (q ^m) middle nonzero element (x ₀, x ₁..., x _255+k-n-1) form 255+k-n interpolation point (x ₀, r ₀), (x ₁, r ₁) ..., (x _255+k-n-1, r _255+k-n-1), then based on (1, k-1)-weighting dictionary inverted sequence table by each interpolation point at least interpolation build a binary polynomial for m time, wherein m is interpolation severe;

S3. utilize the Kotter interpolation algorithm of improvement to ask for minimal polynomial, detailed process is as follows:

S31. first formula is passed through initialization one group of binary polynomial, wherein for initialized jth bar binary polynomial, l _mfor the number of this group binary polynomial, for the set of the binary polynomial composition after initialization, i _kfor iterations, now i _k=0;

S32. formula is passed through cancellation set interior first rank are greater than the binary polynomial of C, wherein $C=n\left(\begin{array}{c}m+1\\ 2\end{array}\right);$

S33. formula is passed through right in the Hasse mixed partial derivative of each binary polynomial calculate, then judge set in the Hasse mixed partial derivative of all binary polynomials whether all equal 0, if all equal 0, then carry out step S36, if be not congruent to 0, carry out step S34;

S34. ask for the minimal polynomial in this group binary polynomial, be shown below:

$f=\underset{j\&Element;J_{i_k}}{\min }{g}_{i_k,j}$

$j^{*}=\arg \underset{j\&Element;J_{i_k}}{\min }{g}_{i_k,j}$

Wherein f is the minimal polynomial in this group binary polynomial, j ^*for the sequence number that minimal polynomial is corresponding;

S35. carry out conversion amendment to the minimal polynomial in this group binary polynomial, formula is as follows:

$g_{i_k+1,j^{*}}={[\mathrm{xf},f]}_{D_{i_k}}={\mathrm{\&Delta;}}_{j^{*}}(x-x_i)\mathrm{fj}=j^{*},$

Remaining binary polynomial also carries out conversion amendment, and formula is as follows:

$g_{i_k+1,j}={[g_{i_k,j},f]}_{D_{i_k}}={\mathrm{\&Delta;}}_{j^{*}}{g}_{i_k,j}-{\mathrm{\&Delta;}}_j\mathrm{fj}\&NotEqual;j^{*};$

S36. the process choosing another group binary polynomial repetition step S31 ~ S35 carries out asking for of this group minimal polynomial, and makes i _k=i _k+ 1;

If S37. i _k=C then stops iteration, now respectively organizes the minimal polynomial g of binary polynomial _c,jcomposition set G _c, by formula Q (x, y)=min{g _c,j| g _c,j∈ G _cminimal polynomial Q (x, y) in C group binary polynomial is solved;

S4. after trying to achieve Q (x, y), Roth-Ruckenstein factoring algorithm is utilized to carry out decomposing message polynomial m'(x corresponding to acquisition RS code Frequency Domain Coding to Q (x, y));

S5. to message polynomial m'(x) adopt Frequency Domain Coding mode to carry out n position coding, get coding codeword n-th-k+1 after obtaining coding codeword and generate message polynomial m (x) corresponding to time domain coding to n bit word, time domain coding mode is adopted to encode to m (x), RS (n, the k) code of correction can be obtained.

2. the matrix two-dimensional code RS decoding error correction method of GS algorithm based on improving according to claim 1, is characterized in that: in step S4, obtains the message polynomial m'(x that RS code Frequency Domain Coding is corresponding) process specific as follows:

After using Roth-Ruckenstein factoring algorithm to decompose Q (x, y), obtain the factor of some shapes as y-p (x), wherein p (x) is polynomial of one indeterminate, decomposes the polynomial of one indeterminate obtained and forms set L:

L={p (x): (y-p (x)) | Q (x, y) and degp (x) < k}

Degp (x) < k represents that the number of times of p (x) is less than k, Frequency Domain Coding mode is adopted to encode to polynomial of one indeterminate all in set L, the code word and RS (n that obtain encoding, k) code compares, and polynomial of one indeterminate corresponding to the minimum code word of Hamming distance is message polynomial m'(x).

3. the matrix two-dimensional code RS decoding error correction method of the GS algorithm based on improving according to claim 1, it is characterized in that: in step S2, the process of the binary polynomial J (x, y) of structure can be represented by the formula:

$\begin{array}{c}J(x,y)=\underset{\mathrm{lod}\left(J(x,y)\right)}{\min }\{J(x,y)\&Element;F_q[x,y\left]\right|D_{\mathrm{\&alpha;\&beta;}}\text{J}(x_i,r_i)=0,i=\mathrm{0,1},...,n-1,\\ \mathrm{\&alpha;}+\mathrm{\&beta;}<m(\mathrm{\&alpha;},\mathrm{\&beta;}\&Element;N)\}\end{array}$

D _{α β}j (x _i, r _i) represent that binary polynomial J (x, y) is at interpolation point (x _i, y _i) (α, β) rank Hasse mixed partial derivative, F _q[x, y] for independent variable be the Polynomial Ring with Two of x and y, F _q[x, y] represents that coefficient belongs to finite field gf (q ^m) in element, F _q[x, y] can be expressed as with formula wherein f _ab∈ GF (q ^m).

4. the matrix two-dimensional code RS decoding error correction method of GS algorithm based on improving according to claim 1, is characterized in that: in step S5, to message polynomial m'(x) to adopt Frequency Domain Coding mode to carry out the process of n position coding specific as follows:

$(c_0,c_1,...,c_{n-1})\stackrel{\mathrm{\&Delta;}}{=}f\left(m^{\&prime;}\left(x\right)\right)=(m\left(1\right),m\left(\mathrm{\&alpha;}\right),\left({\mathrm{\&alpha;}}^2\right),...,m\left({\mathrm{\&alpha;}}^{n-1}\right))$

Wherein $m^{\&prime;}\left(x\right)=m_0+m_{1}x+...+m_{k-1}{x}^{k-1},$ m'(x)∈GF(q ^m)；

(c ₀, c ₁..., c _n-1) be coding codeword, α is finite field gf (q ^m) primitive element.

5. the matrix two-dimensional code RS decoding error correction method of the GS algorithm based on improving according to claim 1, it is characterized in that: in step S5, the process of generator polynomial g (x) can be expressed as follows:

g(x)＝(x-α ^l)(x-α ^l+1)...(x-α ^l+2t-1)

Wherein n-k=2t, l are integer, and α is primitive element, and the generator matrix that m (x) is corresponding can be expressed as:

$G_2=\left[\begin{array}{ccccc}1& 0& ...& 0& x^{n-1}\left(\mathrm{mod}g\left(x\right)\right)\\ 0& 1& ...& 0& x^{n-2}\left(\mathrm{mod}g\left(x\right)\right)\\ \&CenterDot;& \&CenterDot;& & & \&CenterDot;\\ \&CenterDot;& \&CenterDot;& ...& 0& \&CenterDot;\\ \&CenterDot;& \&CenterDot;& & 0& \&CenterDot;\\ 0& 0& ...& 0& x^{n-k}\left(\mathrm{mod}g\left(x\right)\right)\end{array}\right]$

Mod g (x) represents generator polynomial g (x) remainder, obtains result and is error correction codeword polynome;

Codeword polynome c (x) after the above-mentioned coding can obtaining message polynomial m (x) correspondence of through type,

c(x)＝m(x)+[m(x)x ^n-kmodg(x)]

The coefficient of codeword polynome c (x) is the codeword vector of time domain coding, by codeword vector, can obtain RS (n, the k) code of correction.