Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-({text {PD}})-sets of minimum size (s+1) for partial permutation decoding for the binary linear Hadamard code (H_m) of length (2^m), for all (mge 4) and (2 le s le lfloor {frac{2^m}{1+m}}rfloor -1), are constructed. Moreover, recursive constructions to obtain s-({text {PD}})-sets of size (lge s+1) for (H_{m+1}) of length (2^{m+1}), from an s-({text {PD}})-set of the same size for (H_m), are also described. These results are generalized to find s-({text {PD}})-sets for the ({mathbb {Z}}_4)-linear Hadamard codes (H_{gamma , delta }) of length (2^m), (m=gamma +2delta -1), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type (2^gamma 4^delta ). Specifically, s-PD-sets of minimum size (s+1) for (H_{gamma , delta }), for all (delta ge 3) and (2le s le lfloor {frac{2^{2delta -2}}{delta }}rfloor -1), are constructed and recursive constructions are described.