Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes

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1 American Journal of Mathematical and Computer Modelling 2017; 2(1): doi: /.amcm Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes Yani Zhang, Xiaomin Bao, Zhihua Yuan, Xusheng Wu School of Mathematics & Statistics, Southwest University, Chongqing, China address: (Xiaomin Bao) o cite this article: Yani Zhang, Xiaomin Bao, Zhihua Yuan, Xusheng Wu. Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes. American Journal of Mathematical and Computer Modelling. Vol. 2, No. 1, 2017, pp doi: /.amcm Received: October 29, 2016; Accepted: November 28, 2016; Published: January 6, 2017 Abstract: In this paper, a new efficient syndrome-weight decoding algorithm (NESWDA) is presented to decode up to five possible errors in a binary systematic (7,, 11) quadratic residue (QR) code. he main idea of NESWDA is based on the property cyclic codes together with the weight of syndrome difference. he advantage of the NESWDA decoding algorithm over the previous table look-up methods is that it has no need of a look-up table to store the syndromes and their corresponding error patterns in the memory. Moreover, it can be extended to decode all five-error-correcting binary QR codes. Keywords: Cyclic Codes, Decoding, Quadratic Residue Code 1. Introduction he well-known QR codes, introduced by Prange in his report [1] in 197, are cyclic BCH codes with code rates greater than or equal to one-half and generally have large minimum distances so that most of the known QR codes are among the best-known codes. he code augmented by a parity bit, for instance, the (, 12, 8) QR code has be used for numerous communication links, including the Voyager imaging system link of NASA [2]. In the past decades, a series of different decoding methods given in [6-1] have been developed to decode the QR code. However, these algebraic decoding techniques require a large number of complicated computations in a finite field. hese complicated computations will lead to a time delay in the decoding procedures and therefore the decoding time will become unrealistic when the code length is large. herefor, to reduce the decoding complexity, we introduce the NESWDA algorithm, which can be used to decode the (7,,11) QR code. It allows for the correction of up to t = ( d 1) 2 = ( 11 1) 2 = errors, where x denotes the greatest integer less than or equal to x; t is the error-correcting capability and d=11 is the minimum Hamming distance of the code. Recently, table look-up decoding algorithms have been developed to decode the (7,, 11) QR code. he full lookup table in the conventional table look-up decoding algorithm (CLDA) needs = syndromes and their corresponding i error patterns. It requires (6bytes + 3bytes) 1.8 Mbytes memory size to store the table. Such a large memory required makes the computation very complicated. An efficient algorithm named look-up table decoding (LD) algorithm in [3] to decode the (7,, 11) QR code needs 1.0 Mbytes memory size to store the table. he well-known syndrome decoder [2] Requires 1 7 i 7 = = syndromes corresponding to error patterns stored in a table with a (6bytes +3bytes)=10 3. Kbytes memory size, called the reduced lookup table (RL). However, this memory size of the RL is still so large that one needs to further reduce the memory size of the look-up table. o achieve this end, an efficient table look-up decoding algorithm (LDA) in [] is developed to decode the (7,, 11) QR code, the memory size of the developed condensed look-up table consists of 36.6 Kbytes, and Lin et al. [] used a novel table look-up decoding algorithm, called the cyclic weight (CW) decoding algorithm, together with a memory size 20.3 Kbytes to decode the (7,, 11) QR code. As shown in this paper, the proposed NESWDA does not need a memory size to store the look-up table. he prime idea of the proposed

2 American Journal of Mathematical and Computer Modelling 2017; 2(1): NESWDA is based on the weight of syndrome difference between the syndrome of the received word and the row vector of the transpose of the parity-check matrix. Moreover, no complicated computation in the finite field is required in the proposed NESWDA and it also can be extended to decode all five-error-correcting binary QR codes. he remainder of this paper is organized as follows: he background of the binary QR codes is briefly given in Section 2. he proposed NESWDA is described in Section 3. In Section, two examples are used to demonstrate the proposed NESWDA. Finally, this paper concludes with a brief summary in Section. 2. Background of the Binary QR Codes he binary QR codes are a nice family of linear cyclic codes. n, n + 1 2, d denote the binary QR Let ( n, k, d ) or codes with generator polynomial g( x ) over GF ( 2). he length of this code is a prime number of the form n = 8l ± 1, where l is some integer. Also, let k = ( n + 1) 2 denote the message length or information length, and d denote the minimum Hamming distance of the code. he set Q n of quadratic residues modulo n is the set of nonzero squares modulo n ; that is, Q { } 2 n x mod n,1 x n 1 = 7, then its quadratic residue set is =. If n Q 7 = { 1,2,3,,6,78,9,12,1,16,17,18,21,,2,27,28,32,3,36,37,2, }. Let the symbol C 7 denote the binary (7,, 11) QR code. Let α be a root of primitive irreducible polynomial p x = 1+ x + x such that α is a generator of the multiplicative group of all nonzero elements in ( 2 ) i ( β ) i Q7 GF. hen, the element β u = = 178, u = α where 81, is a primitive 7th root of unity in GF ( 2 ). he generator polynomial g ( x ) of the binary (7,, 11) QR code is defined by g x = x = 1+ x + x + x + x + x + x + x + x + x + x + x + x + x + x where the degree of g ( x ) is, which is the multiplicative order of the integer 2 modulo the code length 7; that is, 2 1mod 7. he (7,, 11) QR code generated by this manner can correct up to five errors. A codeword of binary QR code is a polynomial n c x = c + c x + c x + + c x such that it is a multiple n 1 of the generator polynomial g ( x ). If the codeword c x is transmitted through a noisy channel, then the received 2 n 1 polynomial r ( x) = r + r x + r x + + r x can be n 1 expressed as the sum of the codeword polynomial c( x ) and e x = e + e x + e x + + e x. For 2 n 1 the error polynomial n 1 simplicity, let the message or information, codeword, error pattern, received word, and syndrome be expressed as the binary vector forms m = ( m, m,, mk ), c = ( c, c,, cn ), e = ( e0, e1,, en 1 ), r ( r0, r1,, r n 1 ) ( s s ) = and s = 0, 1,, sn k 1, respectively. he systematic codeword of the vector form is given by c = mg, where G is called the k n k matrix and systematic generator matrix. Let A be a I k be a k k identity matrix, and G can be expressed as G =. he parity-check matrix H can be expressed as Ik A k n =, where H A I n k n ( n k) A denotes the ( n k ) k transpose matrix of A. he vector form of the syndrome is defined by s = rh, where n n k transpose matrix of H ; that is, H denotes the H can be expressed as H h0 A h 1 = I = n k h n 1 For C 7, Ahas the following form: Figure 1. A of C Decoding Algorithm and heorems Definition 1. he Hamming weight of a binary vector a is denoted by w(a), and the Hamming distance between a and b (1)

3 8 Yani Zhang et al.: Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes is denoted by d(a, b) = w(a+b). heorem 1. Let a ( a0, a1,, an 1 ) ( b b ) = be two binary vectors, then 0, 1,, bn 1 w(a+b) = w(a) + w(b) = and b n 1 2 aibi. (2) 0 Corollary 1. If aib i = 0 for 0 i n 1, then w(a+b) = w(a) + w(b). (3) he following heorem is useful to compute the syndrome of the received word when the received word shifts one bit to the right. s x be the syndrome polynomial heorem 2. Let corresponding to a received polynomial r ( x ). Also, let r ( 1 ) x be the polynomial obtained by cyclically shifting the coefficients of r ( x) one bit to the right. hen the remainder obtained when dividing xs ( x ) by g ( x ) is the syndrome s ( 1 ) x corresponding to ( 1 r ) ( x ). For a detailed proof, see [2]. However, if the syndrome cyclically shifts many times, then the syndrome computation is quite time-consuming for g x many times. he following dividing xs ( x ) by theorem provides an efficient method to compute s for 0 i n 1, and it can save a lot of computational time. heorem 3. For the binary QR codes, let r be an element of r and h be the th row vector of hen the syndrome s of s H for 0 n 1. r i for 0 i n 1 has the form n 1 = rh[ i+ ], () = 0 where the suffix [ x ] of h denotes x mod n. Proof. Let r ( r0, r1,, r n 1 ) r = ( r,, r, r,, r ) for 0 i n 1 n i n 1 0 n i 1 = and. We have s i = rn ih0 + + rn 1hi 1 + r0 hi + + rn i 1hn 1 = rn ih[ n i i] + + rn 1h[ n 1 i] + r0 h[ 0 i] + + rn i 1h [ n i 1+ i] n 1 = = 0 rh[ i+ ] he proof is thus completed. heorem 3 reveals that the syndrome of can be fast computed by the vector addition. heorem also provides an efficient method to simplify the decoding step by using the syndrome weight. heorem. For the binary QR codes, it is assumed that there are verrors in the received word, where 1 v t and ( d ) t = 1 2. All v errors are in the parity-check bits if r i and only if the weight of syndrome w( s) = v. For a detailed proof, see []. heorem. For the binary QR codes, if v errors are in the information bits of the received word, where 1 v t and ( d ) t = 1 2, then the weight of the corresponding syndrome vector satisfies w( s) d v t + 1 () For a detailed proof, see []. heorem 6. For the binary QR codes, let e = ( e, e,, n ) be an error pattern and e = ( e,, k ), 0 1 e 1 ( e ) parity check section. Assume that ( m ) w( e) t, where t = ( d 1) 2 m 0 e 1 e p = k, en 1 be respectively its message section and w e 1, w( ep ) 1 and, then the weight of the corresponding syndrome vector satisfies Proof. As w ( s) = w( eh ) t 1 w s +. (6) = w( e,0 m 1 H + 01 k,e n k p H ) ( d w( em )) w( e p ) ( ( em ) ( e p )) = d w + w d t t + 1, the proof is thus completed. Given a received word r, the syndrome s of can be fast computed by theorem 3. According to theorem, if 1 w s then error positions are in the parity-check bits of r. If 1 w s, then the error positions are in the information bits of r. Let h denote the th row vector of H, where 0 n 1. Also let z denote the syndrome difference between the syndromes of r and h in each decoding step z. By using the weight of z, the error cases can be quickly determined. Let u 0 = ( 1,0,,0) be a k-tuples unit vector and u i has only one nonzero component at the i th position, where 0 i k 1. By using these properties the decoding algorithm can be constructed. Let case I, C, P denote the error position in the information bits, center bit, and parity-check bits of r, respectively. he decoding steps of the proposed NESWDA work as follows: (1)(No error, P, PP, PPP, PPPP, and PPPPP cases) By 0 w s, theorem 3, compute s and w( s ). If then the information vector is m ( r r ) r i 0 1 r 1 =,,, k. Go to step (8). (2)(I, II, III, IIII, and IIIII cases) By theorem 3, compute s and ( k) w s. If 1 w s, then the corrected

4 American Journal of Mathematical and Computer Modelling 2017; 2(1): information vector is ( ) 0 1 k 1 m = r, r,, r + s,0. Go to step (8). (3)(IP, IPP, IPPP, IPPPP, C, CP, CPP, CPPP, and CPPPP cases) Compute the syndrome difference 3 = s hi for 0 i k 1 w, then the and 3 ). If ( 3 ) corrected information vector m = r0, r1,, r k 1 + u i. Go to step (8). () (IC, IIC, IIIC, IIIIC, IIP, IIIP, and IIIIP cases) Compute the syndrome difference 0 i k 1 and ). If s h i = for w and i = 0, then the corrected information vector m = r, r,, r + u +,0 w and ( ). If 0 1 k 1 k 1 0 i k 1, then the corrected information vector m = r, r,, r +,0. Go to step (8). 0 1 k 1 able 1. he number of error patterns in each decoding step of C 7. Steps Cases Number of error patterns 1 P, PP, PPP, PPPP, PPPPP 1 2 I, II, III, IIII, IIIII 1 3 IP,IPP,IPPP,IPPPP,C,CP,CPP,CPPP,CPPPP IC, IIC, IIIC, IIIIC, IIP, IIIP, IIIIP IIPP, IIPPP, ICP, ICPP, ICPPP IICP, IIICP, IIIPP IICPP 6009 Sum = i ()(IIPP, IIPPP, ICP, ICPP, and ICPPP cases) Compute the syndrome difference = s hi h for 0 i k 1 < and ). If w, then the 3 corrected information vector m = ( r0, r1,, r k 1 ) + u i + u. Go to step (8). (6)(IICP, IIICP, and IIIPP cases) Compute the syndrome difference = s h h for 0 i < k 1 and 6 i 6 ). If 6 ) 3 and i = 0 information vector m = ( r0, r1,, rk 1 ) + uk 1 + ( 6,0) If 6 ) 3 and i > 0 information vector m = ( r, r,, r ) + (,0), then the corrected., then the corrected 0 1 k 1 6. Go to step (8). (7)(IICPP case) Compute the syndrome difference = s h h h for 0 i < k 2 and 7 k 1 i 7 ). If ( 7 ) 2 vector ( ) w, then the corrected information m = r, r,, r k + u i + u + u k Go to step (8). (8) Stop. able 1 lists all the 3 error cases and the number of error patterns in each decoding step of C7 of the proposed NESWDA. he flowchart of the proposed NESWDA is shown in figure 1.. Example In this section, two examples are presented to illustrate the proposed NESWDA. Example 1. Let a information m = ( ) be encoded into a C 7 codeword c = ( ). If the received word r = ( ), then the error pattern e = ( ), which means a III error case. he decoding procedure is composed of the following steps. (1) Compute s = ( ) and w(s) = 12. Since w(s) >, go to step 2. (2) Compute s = ( ) and w( s ) = 3. he corrected information vector is m = ( ) + ( ) = ( ). Go to stop.

5 10 Yani Zhang et al.: Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes Figure 2. Flowchart of the proposed NESWDA.

6 American Journal of Mathematical and Computer Modelling 2017; 2(1): Example 2. Let a information m = ( ) be encoded into a C 7 codeword c = ( ). If the received word r = ( ), then the error pattern e = ( ), which means a IICP case. he decoding procedure is composed of the following steps. (1) Compute s = ( ) and w(s) = 9 >, go to step 2. (2) Compute s = ( ) and w( s ) = 16 >, go to step 3. (3) Compute 3 = s hi for 0 i and 3). ( ) 3 0 = ( ), 3 1 3) = 9. ( ) = ( ), 3 9 3) = 13: ( ) = ( ), 3 3) =. ( ) = ( ), 3) = 11. Since every 3) >, go to step. () Compute = s h for 0 i and ). i = s h0 = ( ) ( ) = ( ), ) = 12. = s h1 = ( ) ( ) = ( ), ) = 6. = s h = ( ) ( ) = ( ), ) = 1. Since every ) >, go to step. () Compute = s hi h for 0 i < and ). = s h0 h = ( ), 1 ) = 6. = s h0 h = ( ), 2 ) = 1. = s h0 h = ( ), w( ) = 11. = s h1 h = ( ), 2 ) = 11. = s h1 h = ( ), ) = 7. = s h22 h = ( ), ) = 10. Since every ) > 3, go to step 6. (6) Compute = s h h for 0 i < and 6) i = s h h = ( ), 6) = 12. = s h h = ( ), 6) = Since 6) = 2 3, and i = 0. he corrected information

7 12 Yani Zhang et al.: Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes Go to stop.. Conclusions ( r r r ) u ( ) m =,,, + +,0 = ( ) + ( ) + ( ) = ( ). A new NESWDA decoding algorithm is developed to correct up to five errors for the approximate half-rate (7,, 11) QR code. he main idea behind the proposed NESWDA is based on the fact that it makes use of the properties of cyclic codes, the weight of syndrome difference. he proposed NESWDA has no need of a look-up table to store the syndromes and their corresponding error patterns in the memory. Moreover, it can be extended to decode all five-error-correcting binary QR codes. References [1] E. Prange, Some Cyclic Error-Correcting Codes with Simple Decoding Algorithms, Air Force Cambridge Research Center, Cambridge, MA, 198 (N-8-16). [2] S. B. Wicker, Error Control Systems for Digital Communication and Storage, Prentice Hall, 199. [3] Y. H. Chen,. K. ruong, C. H. Huang, C. H. Chien, A lookup table decoding of systematic (7,, 11)quadratic residue code, Information Science 179 (2009) []. C. Lin, H. P. Lee, H. C. Chang, S. I. Chu,. K. ruong, High speed decoding of the binary (7,, 11)quadratic residue code, Information Science 180 (2010) []. C. Lin, H. P. Lee, H. C. Chang,. K. ruong, A cyclic weight algorithm of decoding (7,, 11) quadratic residue code, Information Science 197 (2012) [6] G. Dubney, I. S. Reed,. K. ruong, J. Yang, Decoding the (7,, 11) quadratic residue code using bit-error probability estimates, IEEE ransactions on Communications 7 (2009). [7] R. He, I. S. Reed,. K. ruong, X. Chen, Decoding the (7,, 11) quadratic residue code, IEEE ransactions on Information heory 7 (2001) [8]. C. Lin,. K. ruong, and H. P. Lee, Algebraic decoding of the (1, 21, 9) quadratic residue code, Information Sciences 179 (2009) [9] X. Chen, I. S. Reed,. Helleseth, and. K. ruong, Use of Grobner bases to decode binary cyclic codes up to the true minimum distance, IEEE ransactions on Information heory 0 (199) [10] Y. H. Chen,. K. ruong, Y. Chang, C. D. Lee, S. H. Chen, Algebraic decoding of Quadratic Residue codes using Berlekamp Massey algorithm, Journal of Information Science and Engineering (2007) [11] I. S. Reed, X. Chen, Error-Control Coding for Data Networks, Kluwer, Boston, MA, [12] I. S. Reed, M.. Shih,. K. ruong, VLSI design of inverse-free Berlekamp-Massey algorithm, Processings IEE 138 (1991) [13] I. S. Reed,. K. ruong, X. Chen, X. Yin, he algebraic decoding of the (1, 21, 9) Quadratic Residue code, IEEE ransactions on Information heory 38(1992) [1] I. S. Reed, X. Yin,. K. ruong, Algebraic decoding of the (32, 16, 8) Quadratic Residue code, IEEE ransactions on Information heory 36 (1990) [1] I. S. Reed, X. Yin,. K. ruong, J. K. Holmes, Decoding the (, 12, 8) Golay code, IEE Proceedings 137 (3) (1990)

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