Systematic Encoding and Permutation Decoding for Zps-Linear Codes
Adrián Torres-Martín, Mercè Villanueva · IEEE Transactions on Information Theory · 2022
Linear codes over$\mathbb {Z}_{p^{s}}$of length$n$are subgroups of$\mathbb {Z}_{p^{s}}^{n}$. These codes are also called$\mathbb {Z}_{p^{s}}$-additive codes and can be seen as a generalization of linear codes over$\mathbb {Z}_{2}$and$\mathbb {Z}_{4}$. A$\mathbb {Z}_{p^{s}}$-linear code is a code over$\mathbb {Z}_{p}$, not necessarily linear, which is the generalized Gray map image of a$\mathbb {Z}_{p^{s}}$-additive code. In 2015, a systematic encoding was found for$\mathbb {Z}_{4}$-linear codes. Moreover, an alternative permutation decoding method, which is suitable for any binary code (not necessarily linear) with a systematic encoding, was established. In this paper, we generalize these results by presenting a systematic encoding for any$\mathbb {Z}_{p^{s}}$-linear code with$s\geq 2$and$p$prime. We also describe a permutation decoding method for any systematic code over$\mathbb {Z}_{p}$, not necessarily linear, and show some examples of how to use this systematic encoding in this decoding method.