Butson full propelinear codes

TL;DR

This paper explores the structure and transformations of Butson Hadamard matrices and associated codes over finite rings, introducing new morphisms and demonstrating their properties and applications in coding theory.

Contribution

It introduces a new morphism for Butson Hadamard matrices via a generalized Gray map and shows how to construct larger matrices and codes with specific algebraic properties.

Findings

A new morphism for Butson Hadamard matrices is established.

A correspondence between additive codes over inite rings and BH-codes is demonstrated.

Structural properties of cocyclic BH-codes are analyzed.

Abstract

In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of \'{O} Cath\'{a}in and Swartz. That is, we show how, if given a Butson Hadamard matrix over the $k^{\rm th}$ roots of unity, we can construct a larger Butson matrix over the $\ell^{\rm th}$ roots of unity for any $\ell$ dividing $k$, provided that any prime $p$ dividing $k$ also divides $\ell$. We prove that a $\mathbb{Z}_{p^s}$-additive code with $p$ a prime number is isomorphic as a group to a BH-code over $\mathbb{Z}_{p^s}$ and the image of this BH-code under the Gray map is a BH-code over $\mathbb{Z}_p$ (binary Hadamard code for $p=2$). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided.