Systematic Encoding and Permutation Decoding for Zps-Linear Codes

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.