The Zps-additive codes of lengthnare subgroups of Znps, withpprime ands≥ 1. They can be seen as a generalization of linear codes over Z2, Z4, or more general over Z2s. In this paper, we show two methods for computing a parity-check matrix of a Zps-additive code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for linear codes over finite rings in general. Complementing this comparison, we also show a time complexity analysis of the algorithms. The rings Zpsbelong to a more general class of rings: finite chain rings. Along the paper, we observe that the same results can be applied to any linear code over a finite commutative chain ring.

The$\mathbb{Z}_{p^{s}}-\mathbf{additive}$codes of length$n$are subgroups of$\mathbb{Z}_{p^{s}}^{n}$, with$p$prime and$s\geq 1$. They can be seen as a generalization of linear codes over$\mathbb{Z}_{2},\ \mathbb{Z}_{4}$, or more general over$\mathbb{Z}_{2^{s}}$. In this paper, we show two methods for computing a parity-check matrix of a$\mathbb{Z}_{p^{s}} -\mathbf{additive}$code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for codes over finite rings in general.

It is known that$\mathbb {Z}_{p^{s}}$-linear codes, which are the Gray map image of$\mathbb {Z}_{p^{s}}$-additive codes (linear codes over$\mathbb {Z}_{p^{s}}$), are systematic and a systematic encoding has been found. This makes$\mathbb {Z}_{p^{s}}$-linear codes suitable to apply the permutation decoding method, based on the existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. Some constructions of r-PD-sets of minimum size$r+1$for$\mathbb {Z}_{p^{s}}$-linear generalized Hadamard codes of type$(n;t_{1}, {\dots },t_{s})$are known. In this paper, for these codes, we present new constructions of r-PD-sets of size$r+1$, which are suitable for all parameters$t_{1}, {\dots },t_{s}$. These allow us to obtain new r-PD-sets for values of r closer to the theoretical upper bound, improving previous known results.

In a previous work it was shown that ${\mathbb{Z}_{{p^s}}}$-linear codes, which are the Gray map image of ${\mathbb{Z}_{{p^s}}}$-additive codes (linear codes over ${\mathbb{Z}_{{p^s}}}$), are systematic by giving a systematic encoding. This makes ${\mathbb{Z}_{{p^s}}}$-linear codes and, in particular, ${\mathbb{Z}_8}$-linear codes suitable to apply the permutation decoding method. This technique is also based on the existence of s-PD-sets, which are subsets of the permutation automorphism group of the code. In this paper, we study the permutation automorphism group of ${\mathbb{Z}_8}$-linear Hadamard codes of type (n;t1, t2, t3) and show how to find s-PD-sets of minimum size s+1, for all s up to an upper bound, to perform a partial permutation decoding for these 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.