Construction of Hadamard $ {\mathbb {Z}}_{2} {\mathbb {Z}}_{4} {Q}_{8}$ -Codes for Each Allowable Value of the Rank and Dimension of the Kernel
Pere Montolio, Josep Rifà · IEEE Transactions on Information Theory · 2015
This paper deals with Hadamard Z2Z4Q8-codes, which are binary codes after a Gray map from a subgroup of direct products of Z2, Z4, and Q8, where Q8is the quaternionic group. In a previous work, these codes were classified in five shapes. In this paper, we analyze the allowable range of values for the rank and dimension of the kernel, which depends on the particular shape of the code. We show that all these codes can be represented in a standard form, from a set of generators, which can help in understanding the characteristics of each shape. The main results we present are the characterization of Hadamard Z2Z4Q8-codes as a quotient of a semidirect product of Z2Z4-linear codes and the construction of Hadamard Z2Z4Q8-codes with each allowable pair of values for the rank and dimension of the kernel.