Circulant codes based on the prime 29 (Corresp.)

Abstract

Abstract-Somenew examples of binary Bose-ChaudhuriHoquengkn (BCH) codes of length 255 are found for which tbe minimum distance and designed dishace agree.For a cyclic code of length n with generating polynomial g(x), the designed distance S is defined as follows: 6-1 is the maximum length of a string of consecutive powers of a which are zeros of g (x), where a is a primitive nth root of unity. It is known (see e.g. [l, p. 2011) that S is a lower bound on the true minimum distance d. For most Bose-Chaudhuri-Hocquenghen (BCH) codes of primitive length (n x2"--1) we have d= 6, the first example of nonequality being n = 127, 8 -29, d=31 ([ 1, p. 2671).Kasami [2] proved that when m # 8, 12 and m > 6, there exist binary BCH codes with d >I% Thus n = 255 is the first length for which it is not known when d = S or d >I% The aim of this correspondence is to prove that d= 6 for 6 = 37, 39, 45, 87 removing some asterisks from [ 1, p. 2671. Still unsettled values are S =43, 53, 59, 61, 91. We shall make use of the following theorem.Theorem ([3, p. 2781): Let n, and n2 be relatively prime and let a be a factor of ni. If a BCH code of length n2 and designed distance d has minimum distance exactly d, then the BCH code of length nln2 and designed distance ad has minimum distance exactly ad.In [4, table I] we find d= 6 for 9 <S < 29.Case 6 = 37: The (n = 85, k = 9) BCH code, with 8 = 37, admits a codeword of weight 37, namely .This can be checked by "direct" computation. Applying the theorem with n2 = 85, d= 37, n1 = 3, a = 1 yields d= S = 37 for the (255,131) BCH code. Case S= 39: First we construct an (85,8) BCH code with S=40: the roots of its generating polynomial g(x) are the ai, 1 < i < 85 which are not in the cyclotomic class C3 = {3,6,12,24,48,11,22 Inform Theory, vol. IT-19, pp. 818-820, Nov. 1973. -, "Minimum distance bounds for binary linear codes," IEEE Trans.Inform. Theory, vol. IT-19, pp. 344356, May 1973. Circulant Codes Based on the Prime 29 VIJAY K. BHARGAVA, MEMBER, IEEE, AND C. NGUYEN Abstmct-'Ibe weight distributions of an (87,29) and a (58,29) eircuhnt code are en-ted. The clrculants are based on the quadratic residues and the quadratic nooresidues medulo the prime 29. 'fk (87,29) code is known to be cyclic while tbe (5SJ9) code is self-dual. Let A and B denote the (87,29) and (58,29) circulant codes with generator matrix [I,,; Q,N] and [I,,, Q], respectively, where Q and N are the 29x29 circulant matrices based on the quadratic residues and nonresidues, respectively. We omit the details of choosing the first rows of the circulants and their properties as they can be found in [l] and [2]. Circulants based on the quadratic residues and the quadratic nonresidues were first used for code construction by the late Maurice Karlin in a remarkable paper [3]. THE (87,29) CODEThis code is also a cyclic code of the type (3p,p) as described in [4] by MacWilliams. If (u, q,n) is a codeword of the (87,29) code A, so is (u~,q~,mr) where ?r is a cyclic permutation of period 29. Thus the number of codewords of...