On $Z_{p^r}{Z}_{p^r}{Z}_{p^s}$-Additive Cyclic Codes

Abstract

In this paper, we introduce ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes for $r\le s$. These codes can be identified as ${\mathbb{Z}}_{p^s}[x]$-submodules of ${\mathbb{Z}}_{p^r}[x]/⟨x^{\alpha }-1⟩\times {\mathbb{Z}}_{p^r}[x]/⟨x^{\beta }-1⟩\times {\mathbb{Z}}_{p^s}[x]/⟨x^{\gamma }-1⟩$. We determine the generator polynomials and minimal generating sets for this family of codes. Some previous works has been done for the case $p=2$ with $r=s=1$, $r=s=2$, and $r=1,s=2$. However, we show that in these previous works the classification of these codes were incomplete and the statements in this paper complete such classification. We also discuss the structure of separable ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes and determine their generator polynomials. Further, we also study the duality of ${\mathbb{Z}}_{p^s}[x]$-submodules. As applications, we present some examples and construct some optimal binary codes.