Designs, Codes and Cryptography

, Volume 70, Issue 1, pp 105–115

Characterization of the automorphism group of quaternary linear Hadamard codes


DOI: 10.1007/s10623-012-9678-2

Cite this article as:
Pernas, J., Pujol, J. & Villanueva, M. Des. Codes Cryptogr. (2014) 70: 105. doi:10.1007/s10623-012-9678-2


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.


Quaternary linear codes Hadamard codes 1-Perfect codes Permutation automorphism group 

Mathematics Subject Classification

94B25 20B25 

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Departament d’Enginyeria de la Informació i de les ComunicacionsUniversitat Autònoma de BarcelonaBarcelonaSpain