Linear Codes over Galois Ring $GR(p^2,r)$ Related to Gauss sums

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 $Z_4$. In this paper we consider two series of linear codes $C(G)$ and $\widetilde{C}(G)$ over Galois ring $R=GR(p^2,r)$, where $G$ is a subgroup of $R^{(s)^*}$ and $R^{(s)}=GR(p^2,rs)$. We present a general formula on $N_\beta(a)$ in terms of Gauss sums on $R^{(s)}$ for each $a\in R$, where $N_\beta(a)$ is the number of a-component of the codeword $c_\beta\in C(G) (\beta\in R^{(s)})$ (Theorem 3.1). We have determined the complete Hamming weight distribution of $C(G)$ and the minimum Hamming distance of $\widetilde{C}(G)$ for some special G (Theorem 3.3 and 3.4). We show a general formula on homogeneous weight of codewords in $C(G)$ and $\widetilde{C}(G)$ (Theorem 4.5) for the special $G$ given in Theorem 3.4. Finally we obtained series of nonlinear codes over $\mathbb{F}_{q} \ (q=p^r)$ with two Hamming distance by using Gray map (Corollary 4.6).