2016/03/07 by Aixian Zhang, Jin Li, Zhang, Aixian +3
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Computer and information sciences #Finite Group Theory Research #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1603.02018
openalex publication_date 2016/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Linear codes over finite rings become one of hot topics in coding theory after Hommons et al.([4], 1994) discovered that several remarkable nonlinear binary codes with some linear-like properties are the images of Gray map of linear codes over Z4. In this paper we consider two series of linear codes C(G) and \widetildeC(G) over Galois ring R=GR(p2,r), where G is a subgroup of R(s)^* and R(s)=GR(p2,rs). We present a general formula on Nβ(a) in terms of Gauss sums on R(s) for each a∈ R, where Nβ(a) is the number of a-component of the codeword cβ∈ C(G) (β∈ R(s)) (Theorem 3.1). We have determined the complete Hamming weight distribution of C(G) and the minimum Hamming distance of \widetildeC(G) for some special G (Theorem 3.3 and 3.4). We show a general formula on homogeneous weight of codewords in C(G) and \widetildeC(G) (Theorem 4.5) for the special G given in Theorem 3.4. Finally we obtained series of nonlinear codes over \mathbbFq (q=pr) with two Hamming distance by using Gray map (Corollary 4.6).