Abstract
A quaternary linear Hadamard code \({\mathcal{C}}\) is a code over \({\mathbb{Z}_4}\) such that, under the Gray map, gives a binary Hadamard code. The permutation automorphism group of a quaternary linear code \({\mathcal{C}}\) of length n is defined as \({{\rm PAut}(\mathcal{C}) = \{\sigma \in S_{n} : \sigma(\mathcal{C}) = \mathcal{C}\}}\). In this paper, the order of the permutation automorphism group of a family of quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by computing the orbits of the action of \({{\rm PAut}(\mathcal{C})}\) on \({\mathcal{C}}\) and by giving the generators of the group. Since the dual of a Hadamard code is an extended 1-perfect code in the quaternary sense, the permutation automorphism group of these codes is also computed.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding Theory and Applications”.
The material in this paper was presented in part at the 3rd International Castle Meeting on Coding Theory and Applications, Cardona, Spain, 11–15 Sep 2011 [1].
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Pernas, J., Pujol, J. & Villanueva, M. Characterization of the automorphism group of quaternary linear Hadamard codes. Des. Codes Cryptogr. 70, 105–115 (2014). https://doi.org/10.1007/s10623-012-9678-2
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DOI: https://doi.org/10.1007/s10623-012-9678-2