@article{article_514339, title={Codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$}, journal={Journal of Algebra Combinatorics Discrete Structures and Applications}, volume={6}, pages={39–51}, year={2019}, DOI={10.13069/jacodesmath.514339}, author={Aydogdu, İsmail}, keywords={Linear codes,Self-dual codes,$\mathbb{Z}_{2}\mathbb{Z}_{2}(u)$-linear codes,$\mathbb{Z}_{p}(u^ru^s)$-linear codes}, abstract={In this paper we generalize $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes to codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes as $\mathbb{Z}_{p}[u^r,u^s]$-linear codes which are actually special submodules. We determine the standard forms of the generator and parity-check matrices of these codes. Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $\mathbb{Z}_{2}[u^r,u^s]$-linear codes. Finally, we study the structure of self-dual $\mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.}, number={1}, publisher={iPeak Academy}