Partial permutation decoding for ℤ8-linear Hadamard codes

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;t
1
, t
2
, t
3
) 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.