Abstract
It is known that there are exactly \(\lfloor \frac{t-1} {2} \rfloor\) and \(\lfloor \frac{t} {2}\rfloor\) nonequivalent \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear Hadamard codes of length 2t, with α = 0 and \(\alpha \not =0\), respectively, for all t ≥ 3. In this paper, it is shown that each \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear Hadamard code with α = 0 is equivalent to a \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear Hadamard code with α ≠ 0, so there are only \(\lfloor \frac{t} {2}\rfloor\) nonequivalent \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear Hadamard codes of length 2t. Moreover, the orders of the permutation automorphism groups of the \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear Hadamard codes are given.
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Acknowledgements
The work of the first author has been partially supported by the Russian Foundation for Basic Research under Grant 13-01-00463-a and by the Target Program of SB RAS for 2012-2014 (integration project no. 14).
The work of the second author has been partially supported by the Spanish MICINN under Grants TIN2010-17358 and TIN2013-40524-P and by the Catalan AGAUR under Grant 2014SGR-691.
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Krotov, D.S., Villanueva, M. (2015). On the Automorphism Groups of the \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-Linear Hadamard Codes and Their Classification. In: Pinto, R., Rocha Malonek, P., Vettori, P. (eds) Coding Theory and Applications. CIM Series in Mathematical Sciences, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-17296-5_25
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DOI: https://doi.org/10.1007/978-3-319-17296-5_25
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