$SR$-Additive cyclic codes are asymptotically good

Abstract

Recently, Yao et al. \cite{ZH} showed that $Z_{p}Z_{p}[u]/\langle u^{t}\rangle $-additive cyclic codes are asymptotically good. We extend the study to the double cyclic codes over two finite commutative chain rings $S=Z_{p}[u]/ \langle u^s\rangle$ and $ R=Z_{p}[u]/ \langle u^r\rangle$ with $ 1\leq s < r $. For positive integers $\alpha$ and $\beta$, define $S_{\alpha}=\frac{S[x]}{\langle x^{\alpha}-1\rangle}$ and $R_{\beta}=\frac{R[x]}{\langle x^{\beta}-1\rangle}$. A subset $C\subseteq S_{\alpha}\times R_{\beta}$ is called an $SR$-additive cyclic code if $C$ is an $R[x]$-submodule. In this paper, we show that $SR$-additive cyclic codes are asymptotically good. To show this, we construct a class of $ SR$ -additive cyclic codes generated by pairs of polynomials. Based on probabilistic methods and Gilbert-Varshamov bound, we determine the asymptotic rates and relative minimum distances of this class of codes. In particular, we prove that there exist numerous asymptotically good $SR$-additive cyclic codes with the relative minimum distance of the code is convergent to $\delta$, and the rate is convergent to $ \frac{1}{1+p^{r-s}} $ for $ 0 \leq \delta \leq \frac{1}{1+p^{r-s}}$ and $ 1\leq s < r $.