Partial permutation decoding for binary linear and Z4-linear Hadamard codes

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Permutation decoding is a technique which involves finding a subset S, called PD-set, of the permutation automorphism group of a code C in order to assist in decoding. An explicit construction of 2ᵐ-m-1/1+m -PD-sets of minimum size 2ᵐ-m-1/1+m + 1 for partial permutation decoding for binary linear Hadamard codes Hₘ of length 2ᵐ, for all m≥ 4, is described. Moreover, a recursive construction to obtain s-PD-sets of size l for Hₘ₊₁ of length 2ᵐ⁺¹, from a given s-PD-set of the same size for Hₘ, is also established. These results are generalized to find s-PD-sets for (nonlinear) binary Hadamard codes of length 2ᵐ, called Z₄-linear Hadamard codes, which are obtained as the Gray map image of quaternary linear codes of length 2ᵐ⁻¹.