3 years ago

Classification of the Z2Z4-linear Hadamard codes and their automorphism groups.

Denis Krotov, Mercè Villanueva

A $Z_2Z_4$-linear Hadamard code of length $\alpha+2\beta=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $\alpha$ binary coordinates and $\beta$ quaternary coordinates. It is known that there are exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $\alpha=0$ and $\alpha\not=0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $\alpha=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $\alpha\not=0$; so there are only $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2Z_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2Z_4$-linear Hadamard codes are given.

Publisher URL: http://arxiv.org/abs/1408.1147

DOI: arXiv:1408.1147v2

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