Algebraic decoding of (71, 36, 11), (79, 40, 15), and (97, 49, 15) quadratic residue codes

Recently, a new algebraic decoding algorithm for quadratic residue (QR) codes was proposed by Truong et al. Using that decoding scheme, we now develop three decoders for the QR codes with parameters (71, 36, 11), (79, 40, 15), and (97, 49, 15), which have not been decoded before. To confirm our results, an exhaustive computer simulation has been executed successfully.     

Algebraic Decoding of the $(89, 45, 17)$ Quadratic Residue Code

Recently, an algebraic decoding algorithm suggested by Truong (2005) for some quadratic residue codes with irreducible generating polynomials has been designed that uses the inverse-free Berlekamp–Massey (BM) algorithm to determine the error-locator polynomial. In this paper, based on the ideas of the algorithm mentioned above, an algebraic decoder for the $(89, 45, 17)$ binary quadratic residue code, the last one not decoded yet of length less than $100$ , is proposed. It was also verified theoretically for all error patterns within the error-correcting capacity of the code. Moreover, the verification method developed in this paper can be extended for all cyclic codes without checking all error patterns by computer simulations.      

The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code

An extremal self-dual doubly-even binary (n,k,d) code has a minimum weight d=4/spl lfloor/n/24/spl rfloor/+4. Of such codes with length divisible by 24, the Golay code is the only (24,12,8) code, the extended quadratic residue code is the only known (48,24,12) code, and there is no known (72,36,16) code. One may partition the search for a (48,24,12) self-dual doubly-even code into three cases. A previous search assuming one of the cases found only the extended quadratic residue code. We examine the remaining two cases. Separate searches assuming each of the remaining cases found no codes and thus the extended quadratic residue code is the only doubly-even self-dual (48,24,12) code.     

A self-dual even (96, 48, 16) code (Corresp.)

IfC_1, C_2are two codes of the same length, letC_1 neq C_2 = {(alpha + eta,beta + eta, alpha + beta + eta) mid alpha, beta in C_1, eta in C_2}. TakingC_1andC_2to be two extended quadratic residue codes of length 32, it is shown thatC_1 neq C_2is a (96,48,16) self-dual even code with all weights divisible by 4.     

The algebraic decoding of the (41, 21, 9) quadratic residue code

A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code, is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic, and quartic equations over GF(2^m) in a new way which uses Zech's logarithms for the arithmetic. The logarithms developed for Zech's logarithms save a substantial amount of computer memory by storing only a table of Zech's logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed can apply generally to other codes such as the BCH and Reed-Solomon codes     

Algebraic decoding of the (32, 16, 8) quadratic residue code

An algebraic decoding algorithm for the 1/2-rate (32, 16, 8) quadratic residue (QR) code is found. The key idea of this algorithm is to find the error locator polynomial by a systematic use of the Newton identities associated with the code syndromes. The techniques developed extend the algebraic decoding algorithm found recently for the (32, 16, 8) QR code. It is expected that the algebraic approach developed here and by M. Elia (1987) applies also to longer QR codes and other BCH-type codes that are not fully decoded by the standard BCH decoding algorithm     

Cyclotomy and duadic codes of prime lengths

We present a cyclotomic approach to the construction of all binary duadic codes of prime lengths. We calculate the number of all binary duadic codes for a given prime length and that of all duadic codes that are not quadratic residue codes. We give necessary and sufficient conditions for p such that all binary duadic codes of length p are quadratic residue (QR) codes. We also show how to determine some weights of duadic codes with the help of cyclotomic numbers     

On the Pless-construction and ML decoding of the (48,24,12) quadratic residue code

We present a method for maximum likelihood decoding of the (48,24,12) quadratic residue code. This method is based on projecting the code onto a subcode with an acyclic Tanner graph, and representing the set of coset leaders by a trellis diagram. This results in a two level coset decoding which can be considered a systematic generalization of the Wagner rule. We show that unlike the (24,12,8) Golay code, the (48,24,12) code does not have a Pless-construction which has been an open question in the literature. It is determined that the highest minimum distance of a (48,24) binary code having a Pless (1986) construction is 10, and up to equivalence there are three such codes.     

A note on extended quadratic residue codes overGF(9)and their ternary images (Corresp.)

We discuss[2(p + 1), p + 1]double circulant codes which are the ternary images of the[p + 1,(p + 1)/2]extended quadratic residue codes over GF(9). Herepis a prime of the formp = 12k pm 5. As a special result we obtain a[64, 32,18]ternary self-dual code which is the largest known code meeting the bound of Mallows and Sloane.     

The weight distributions of some binary quadratic residue codes

The weight distributions of binary quadratic residue codes C can be computed from the weight distribution of a subset of C containing one-fourth (resp., one-eighth) of the codewords in C when the length of the code is congruent to 1 (resp., -1) modulo 8. An algorithm to determine the weight distributions of binary cyclic codes is given. As a consequence, the weight distributions of (73,37,13), (89,45,17), and (97,49,15) quadratic residue codes are determined precisely.     

Split group codes

We construct a class of codes of length n such that the minimum distance d outside of a certain subcode is, up to a constant factor, bounded below by the square root of n, a well-known property of quadratic residue codes. The construction, using the group algebra of an Abelian group and a special partition or splitting of the group, yields quadratic residue codes, duadic codes, and their generalizations as special cases. We show that most of the special properties of these codes have analogues for split group codes, and present examples of new classes of codes obtained by this construction     

Hard- and soft-decision decoding beyond the half minimum distance---An algorithm for linear codes (Corresp.)

A decoding algorithm for linear codes that uses the minimum weight words of the dual code as parity checks is defined. This algorithm is able to correct beyond the half minimum distance and has the capability of including soft-decision decoding. Results on applying this algorithm to quadratic residue (QR) codes, BCH codes, and the Golay codes (with and without soft-decision decoding) are presented.     

A New Scheme to Determine the Weight Distributions of Binary Extended Quadratic Residue Codes

This letter proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleason?s theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distributions of binary (138, 69, 22) and (168, 84, 24) extended quadratic residue codes are given.     

A double circulant presentation for quadratic residue codes (Corresp.)

Forp eqiv pm 1 pmod{8}there are two binary codes,Q(p)andN(p), each an extended quadratic residue code of lengthp+1and dimension(p+1)/2. The existence of double circulant generator matrices for these codes is investigated. A possibly infinite family of primespis presented for whichQ(p)andN(p)must have double circulant generator matrices. Two counterexamples prove the construction is not always possible.     

Generalized quadratic residue codes

A simple definition of generalized quadratic residue codes, that is, quadratic residue codes of block lengthp^{m}, is given, and an account of many of their properties is presented.     

The automorphism groups of the generalized quadratic residue codes

We give the full automorphism groups as groups of semiaffine transformations, of the extended generalized quadratic residue codes. We also present a proof of the Gleason-Prange theorem for the extended generalized quadratic residue codes that relies only on their definition and elementary theory of linear characters     

Universal decoding algorithm for binary quadratic residue codes using syndrome-weight determination

A novel decoding scheme, called syndrome-weight determination, was proposed by Chang et al. in 2008 for the Golay code, or the (23, 12, 7) quadratic residue code. This method is not only very simple in principle but also suitable for parallel hardware design. Presented is a modified version for any binary quadratic residue codes which has been developed. Because of its regular property, the proposed decoder is suitable for both software design and hardware development.     

23 does not divide the order of the group of a (72,36,16) doubly even code (Corresp.)

An interesting open question is whether a(72, 36, 16)doubly even codeCexists. In [3] the odd prime numbers which can divide the order of the group ofCwere determined and23is the largest of these. Twenty-three is eliminated by reducing the problem to the consideration of348codes, each of which is shown to have minimum weight12or less. One of these codes, denoted byC', arises from the(a + x, b + x, a + b + x)construction whereaandbare in one quadratic residue code andxis in the other. The weight distribution ofC'is given.     

Two quaternary quadratic residue codes

The weight distributions of the (13, 6) and the (17, 8) quaternary quadratic residue codes are computed.     

A Unified Method for Determining the Weight Enumerators of Binary Extended Quadratic Residue Codes

This letter proposes an improved and unified method to determine the weight enumerators of binary extended quadratic residue (EQR) codes. It is faster than the previous methods for some of binary EQR codes. Moreover, all the results for the weight enumerators of binary EQR codes are listed.