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Permutation decoding is a technique which involves finding a subset S, called PDset, of the permutation automorphism group PAut(C) of a code C in order to assist in decoding. A method to obtain s-PD-sets of size s + 1 for partial permutation decoding for the binary linear Hadamard codes H m of length 2 m , for all m ≥ 4 and 1 < s ≤ (2 m − m − 1)/(1 + m) , is described. Moreover, a recursive construction to obtain s-PD-sets of size s + 1 for H m+1 of length 2 m+1 , from a given s-PD-set of the same size for the Hadamard code of half length H m is also established.

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In this paper, s-PD-sets of minimum size s + 1 for partial permutation decoding for the binary linear Hadamard code Hm of length 2 m , for all m ≥ 4 and 2 ≤ s ≤ | 2 J 1, are constructed. Moreover, recursive constructions to obtain s-PD-sets of size l ≥ s + 1 for Hm+1 of length 2 m+1 , from an s-PD-set of the same size for Hm, are also described. These results are generalized to find s-PD-sets for the Z4-linear Hadamard codes Hγ,δ of length 2 m , m = γ + 2δ 1, which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type 2 γ 4 δ . Specifically, s-PD-sets of minimum size s + 1 for Hγ,δ , for all δ ≥ 3 and 2 ≤ s ≤ | 2 J1, are constructed and recursive constructions are described.Keywords automorphism group · permutation decoding · PD-set · Hadamard code · Z4-linear code Mathematics Subject Classiftcation (2010) 94B05 · 94B35 · 94B60 IntroductionA binary Hadamard code of length n is a binary code with 2n codewords and minimum distance n/2 [16, Ch.2. §3.]. It is well known that there is a unique binary linear Hadamard code Hm of length n = 2 m , for any m ≥ 2, which is the dual of the extended Hamming code of length 2 m and also coincides with the first order Reed-Muller code of the same length [16, Ch.13. §3]. Binary

Permutation decoding is a technique which involves finding a subset S, called PDset, of the permutation automorphism group of a code C. Constructions of small PD-sets for partial decoding for two families of Z 4-linear codes (Hadamard and Kerdock) are given. Moreover, different decoding methods for Z 4-linear codes are compared by showing their performance applied to these two families.