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

In this paper, we introduce $\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes for $r\leq s$. These codes can be identified as $\mathbb{Z}_{p^s}[x]$-submodules of $\mathbb{Z}_{p^r}[x]/\langle x^{\alpha}-1\rangle \times \mathbb{Z}_{p^r}[x]/\langle x^{\beta}-1\rangle\times \mathbb{Z}_{p^s}[x]/\langle x^{\gamma}-1\rangle$. 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.