A Viewpoint on the Decoding of the Quadratic Residue Code of Length 89

A viewpoint on the weight-6 error patterns of the algebraic decoding of the (89, 45, 17) quadratic residue (QR) code with reducible generator polynomial, proposed by Truong et al. (2008), is presented in this paper. Some weight-6 error patterns will cause a zero value in the syndrome S1. However, in this case, the inverse-free Berlekamp-Massey (IFBM) algorithm is still valid to determine the error-locator polynomial of six errors in the finite field GF(2). An example dem-


Introduction
The famous QR codes, introduced by Prange [1] in 1957, are cyclic BCH codes with code rates greater than or equal to one-half. In addition, the codes generally have large minimum distances so that most of the known QR codes are the best-known codes. In the past decades, several decoding skills have been developed to decode the binary QR codes. The ADAs most used to decode the QR codes are the Newton identities with either Sylvester resultants [3][4][5][6][10][11] or Gröbner bases [13], or IFBM algorithm [7][8][9]12] to determine the error-locator polynomial. The ADA of the (89, 45, 17) QR code [9,12] can correct up to eight errors in GF (2 11 ), because the error-correcting capability of the code is x denotes the greatest integer less than or equal to x, and d = 17 is the minimum Hamming distance of the code. In each decoding procedure, the IFBM algorithm [14] is used to determine the error-locator polynomial of the received sequence [9,12]. Finally, the Chien algorithm [15] is applied to find the roots of the error-locator polynomial. For the QR codes with irreducible generator polynomial, syndrome S 1 = 0 means that the received word has no errors. However, for the (89, 45, 17) QR codes, the syndrome S 1 = 0 means that S 2 , S 4 , …, are all equal to zero and the zero S 1 does not cause a decoding failure in decoding weight-6 error patterns while using the IFBM algorithm to determine the error-locator polynomial.
The rest parts of the paper are organized as follows: The background of (89, 45, 17) QR codes is briefly given in Section 2. The discussion of the weight-6 error patterns is

Background of the Binary (89, 45, 17) QR Code
The codeword of the binary (n, k, d) QR code is defined algebraically as a multiple of its generator polynomial g(x) with coefficients in GF (2). Let the length of the code n be a prime number of the form n = 8m ± 1, where m is a positive integer and m be the smallest positive integer such that 2 m ≡ 1 (mod n). Thus, GF(2 m ) is the extension field of GF (2). Also, let k = (n+1)/2 be the message length and d be the minimum Hamming distance or Hamming weight of the code. The generator polynomial as a cyclic code is given by , where the element β is a primitive nth root of unity in GF (2 m ) and Q n denotes the set of quadratic residues given by Q n = {j|j ≡ x 2 mod n for 1 ≤ x ≤ (n-1)/2}. The set Q n can thus be represented as a disjoint union of cyclotomic cosets, modulo n. These cyclotomic cosets are defined as Q r = {r2 j |j = 0, 1, … , n r -1}, where n r is the smallest positive integer such that n r r r n mod 2 ≡ , n r divides (n-1)/2, and r is the smallest element in Q r . The element r is called the representative element of the cyclotomic cosets Q r . The set S, consisting of all representatives of the QR code, is called the base set of the QR code. These definitions and properties cause the equality  S r r n Q Q ∈ = relating Q n to the cyclotomic cosets, modulo n. Let an element α ∈ GF(2 11 ) be a root of the primitive polynomial p(x) = x 11 + x 2 + 1. Then, α generates the multiplicative group of nonzero elements in GF (2 11 ). Also, let an element β = α u , where u = (2 m -1)/n = (2 11 -1)/89 = 23, is a primitive 89th root of unity in GF (2 11 ); that is, β = α 23 . The base set of this code is S = {1, 3,5,9,11,13,19, 33} and r ∈ S. Therefore, the eight cyclotomic cosets Q r are shown in Table 1.
The minimal polynomial g r (x) can also be expressed as consists of four minimum polynomials given below: (   2  3  5  7  10  1  1  13   14  15  16  18  19  20  21  22   23  24  25  26  28  29  30  31   33  34  37  39  41  42 Since the codeword of the QR code is a multiple of the g(x), the codeword polynomial of the QR code of length 89 can be represented by , where C i ∈ GF(2) for 0 ≤ i ≤ 88, and m(x) = m 44 x 44 +  + m 1 x + m 0 denotes information polynomial, where m i ∈ GF(2) for 0 ≤ i ≤ 44. In such a representation, this type of codeword is called the non-systematic encoding. In practice, the encoding procedure is often implemented by the use of systematic encoding. Let p(x) = p 43 x 43 +  + p 1 x + p 0 be the parity-check polynomial, where p i ∈ GF(2) for 0 ≤ i ≤ 43. Also, let m(x)x n-k divide by g(x), then we get the following identity: Multiplying both sides of (4) by x k and using x n = 1, Then, which is a multiple of g(x), has m(x) in its lower k bits and p(x) = d(x)x k in its higher n -k bits. Thus, the codeword can be represented by the equation below.
As shown in (5), this form of the codeword is called systematic encoding. Now, let a codeword be transmitted through an additive white Gaussian noise (AWGN) channel, and obtain a received word of the form r(x) = c(x) + e(x), where e(x) = e 88 x 88 ++ e 1 x + e 0 is the error polynomial and e i ∈ GF (2). For simplification, the polynomial form can be expressed as the vector form.
β β (6) where i (mod 89) ∈ Q 89 . If i ∉ Q 89 , the syndromes are called unknown syndromes. All the known and unknown syndromes can be expressed as some powers of S 1 , S 5 , S 9 , S 11 , and S 3 , S 13 , S 19 , S 33 , called the primary known and unknown syndromes, respectively. Notice that S 0 = 0 or 1 depends on the fact that v is even or odd, where v ≤ t is the actual number of errors to be corrected and 1 ≤ v ≤ 8.
If there are v errors in r(x), then e(x) has v nonzero terms over GF (2); that is, v r r r , where 0 ≤ r 1 < r 2 < ... < r v ≤ n-1. For i ∈ Q n , the syndrome can be written as . Assuming that v errors occur, the error-locator polynomial L v (x) is defined by where j r j β X = are called the error locators, the σ j are called the elementary symmetric functions for 1 ≤ j ≤ v, and σ 0 = 1. The roots of L v (x) are the inverse of the v error locators {X j }.
To determine the error-locator polynomial, the steps of the IFBM algorithm [14] are summarized as below: where 5. Set k = k + 1. If k ≤ 2t, then go to 2. Otherwise stop.

Discussion of the Weight-6 Error Patterns
The ADA given in [9,12] utilized the IFBM algorithm to determine the error-locator polynomial. In order to apply the IFBM algorithm, the consecutive syndromes S i for 1 ≤ i ≤ 16 need to be computed. Among them, the unknown syndromes S 3 , S 6 , S 7 , S 12 , S 13 , S 14 , and S 15 can be expressed as some powers of the primary unknown syndrome S 3 or S 13 that cannot be computed directly from the received word. The determination of the S 3 and S 13 is given in [9,12].
For the (89, 45, 17) QR code, a C++ program shows that over 200,000 weight-6 error patterns will cause a zero S 1 . However, the zero S 1 does not cause a decoding failure while using the IFBM algorithm to determine the error-locator polynomial. The following example demonstrates the fact.