Partial permutation decoding for binary linear and $$Z_4$$ 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.