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

Abstract

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 $m\ge 4$ 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 $l\ge s+1$ 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 $2^m$, $m=\gamma +2\delta -1$, 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 $2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1$, are constructed and recursive constructions are described.