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

Abstract

A $Z_2{Z}_4$-linear Hadamard code of length $\alpha +2\beta =2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2{Z}_4$-additive code with $\alpha$ binary coordinates and $\beta$ quaternary coordinates. It is known that there are exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2{Z}_4$-linear Hadamard codes of length $2^t$, with $\alpha =0$ and $\alpha \ne 0$, respectively, for all $t\ge 3$. In this paper, it is shown that each $Z_2{Z}_4$-linear Hadamard code with $\alpha =0$ is equivalent to a $Z_2{Z}_4$-linear Hadamard code with $\alpha \ne 0$; so there are only $[t/2]$ nonequivalent $Z_2{Z}_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2{Z}_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2{Z}_4$-linear Hadamard codes are given.

v2: missed overdots in Th.2 corrected, cosmetic changes

Cited by in corpus (1)

• On the Automorphism Groups of the Z2Z4-Linear 1-Perfect and Preparata-Like Codes