Hadamard Z2Z4Q8-codes. Constructions based on the rank and dimension of the kernel

TL;DR

This paper classifies Hadamard Z2Z4Q8-codes based on their shape, analyzes the range of rank and kernel dimension, and provides a standard form and construction methods for these codes.

Contribution

It introduces a standard form for Hadamard Z2Z4Q8-codes and characterizes them as quotients of semidirect products, enabling construction with specific rank and kernel dimensions.

Findings

All codes can be represented in a standard form from a set of generators.

Hadamard Z2Z4Q8-codes are characterized as quotients of semidirect products.

Constructive methods for codes with any allowable rank and kernel dimension.

Abstract

This work deals with Hadamard Z2Z4Q8-codes, which are binary codes after a Gray map from a subgroup of the direct product of Z2, Z4 and Q8 groups, where Q8 is the quaternionic group. In a previous work, these kind of 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 help to a well understanding of the characteristics of each shape. The main results are the characterization of Hadamard Z2Z4Q8-codes as a quotient of a semidirect product of Z2Z4-linear codes and, on the other hand, the construction of Hadamard Z2Z4Q8-codes code with any given pair of allowable parameters for the rank and dimension of the kernel.