On the permutation decoding for binary linear and Z4-linear Hadamard codes

Permutation decoding is a technique, introduced in [3] by MacWilliams, that strongly depends on the existence of special subsets, called PD-sets, of the permutation automorphism group PAut(C) of a linear code C. In [2], it is shown how to find s-PD-sets of minimum size s + 1 for partial permutation decoding for the binary simplex code Sm of length 2 − 1, for all m ≥ 4 and 1 < s ≤ ⌊ 2−m−1 m ⌋ . In [1], an alternative permutation decoding method is presented, which can be applied to any binary systematic code (not necessarily linear), in particular to any Z4-linear code. Nevertheless, this alternative method assumes that we know an appropriate PD-set for such codes. In this talk, we obtain s-PD-sets of size s+1 for binary linear Hadamard codes (extended codes of Sm), following the techniques described in [2]. Furthermore, we provide a criterion to obtain s-PD-sets of the same size for partial permutation decoding for Z4-linear codes. As particular examples, we apply this criterion to (nonlinear) Hadamard Z4-linear codes, where we also prove that such sets are of minimum size. Finally, we present two recursive constructions to obtain s-PD-set for this family of Hadamard Z4-linear codes.