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

TL;DR

This paper classifies Z2Z4-linear Hadamard codes of length 2^t, showing equivalence between certain classes and determining their automorphism groups, thus advancing understanding of their structure.

Contribution

It proves that all Z2Z4-linear Hadamard codes with alpha=0 are equivalent to those with alpha≠0, reducing the classification to [t/2] classes, and computes their automorphism groups.

Findings

Number of nonequivalent codes is [t/2]

All codes with alpha=0 are equivalent to those with alpha≠0

Automorphism groups' orders are explicitly given

Abstract

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.