Improving Explicit Constructions of r-PD-Sets for Zₚs-Linear Generalized Hadamard Codes

Abstract

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.