Improving Explicit Constructions of <i>r</i>-PD-Sets for <i>Zₚs</i>-Linear Generalized Hadamard Codes

It is known that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>-linear codes, which are the Gray map image of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>-additive codes (linear codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>), are systematic and a systematic encoding has been found. This makes <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>-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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r+1$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>-linear generalized Hadamard codes of type <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n;t_{1}, {\dots },t_{s})$ </tex-math></inline-formula> are known. In this paper, for these codes, we present new constructions of r-PD-sets of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r+1$ </tex-math></inline-formula>, which are suitable for all parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t_{1}, {\dots },t_{s}$ </tex-math></inline-formula>. These allow us to obtain new r-PD-sets for values of r closer to the theoretical upper bound, improving previous known results.