On the Automorphism Groups of the $$\mathbb{Z}_{2}\mathbb{Z}_{4}$$ -Linear Hadamard Codes and Their Classification

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 2 t , 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 2 t . Moreover, the orders of the permutation automorphism groups of the \(\mathbb{Z}_{2}\mathbb{Z}_{4}\)-linear Hadamard codes are given.