ONE-HOMOGENEOUS WEIGHT CODES OVER FINITE CHAIN RINGS

Abstract

Abstract. This paper determines the structures of one-homogeneousweight codes over finite chain rings and studies the algebraic propertiesof these codes. We present explicit constructions of one-homogeneousweight codes over finite chain rings. By taking advantage of the distance-preserving Gray map defined in [7] from the finite chain ring to its residuefield, we obtain a family of optimal one-Hamming weight codes over theresidue field. Further, we propose a generalized method that also includesthe examples of optimal codes obtained by Shi et al. in [17]. 1. IntroductionConstant-weight codes represent an important class of codes within the fam-ily of error-correcting codes [11]. A linear code having constant-weight meansthat every nonzero codeword has the same weight. In the literature there aremany papers on binary constant-weight codes which have several applicationssuch as the design of demultiplexers for nano-scale memories [8] and the con-struction of frequency hopping lists for use in GSM networks [12]. Especially,considerable research has been done on the central problem regarding constant-weight codes which is the determination of A(n,d,w), the largest possible sizeof a constant-weight code of length n, Hamming distance at least d, and con-stant weight w. Due to the difficulty in finding good constant-weight codes,various upper and lower bounds on A(n,d,w) have been developed [1, 3, 15, 18].Moreover, there are further studies in this direction including nonbinary finitefields in [2, 10]. It has been shown that there exists a unique one-weight binarylinear code of dimension k such that any two columns in its generator matrixare linearly independent for every positive integer k. Later, this result has beenextended to the ring Z