In this paper, s- $${\text {PD}}$$ PD -sets of minimum size $$s+1$$ s + 1 for partial permutation decoding for the binary linear Hadamard code $$H_m$$ H m of length $$2^m$$ 2 m , for all $$m\ge 4$$ m ≥ 4 and $$2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1$$ 2 ≤ s ≤ ⌊ 2 m 1 + m ⌋ - 1 , are constructed. Moreover, recursive constructions to obtain s- $${\text {PD}}$$ PD -sets of size $$l\ge s+1$$ l ≥ s + 1 for $$H_{m+1}$$ H m + 1 of length $$2^{m+1}$$ 2 m + 1 , from an s- $${\text {PD}}$$ PD -set of the same size for $$H_m$$ H m , are also described. These results are generalized to find s- $${\text {PD}}$$ PD -sets for the $${\mathbb {Z}}_4$$ Z 4 -linear Hadamard codes $$H_{\gamma , \delta }$$ H γ , δ of length $$2^m$$ 2 m , $$m=\gamma +2\delta -1$$ m = γ + 2 δ - 1 , which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type $$2^\gamma 4^\delta $$ 2 γ 4 δ . Specifically, s-PD-sets of minimum size $$s+1$$ s + 1 for $$H_{\gamma , \delta }$$ H γ , δ , for all $$\delta \ge 3$$ δ ≥ 3 and $$2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1$$ 2 ≤ s ≤ ⌊ 2 2 δ - 2 δ ⌋ - 1 , are constructed and recursive constructions are described.
Designs, Codes and Cryptography – Springer Journals
Published: Feb 20, 2017
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