Necessary Conditions for the Existence of Group-Invariant Butson Matrices and a New Family of Perfect Arrays

TL;DR

This paper investigates the conditions under which group-invariant Butson matrices exist, focusing on specific group types, and introduces a new family of perfect arrays derived from these matrices.

Contribution

It provides necessary conditions for the existence of $BH(G,h)$ matrices for certain groups and constructs a new family of perfect arrays from these matrices.

Findings

2687 open cases for $BH( ext{Z}_n,h)$ existence with $1\leq n,h \leq 100$

Established relations between group structure and matrix existence

Constructed new perfect polyphase arrays from Butson matrices

Abstract

Let $G$ be a finite abelian group and let $\exp(G)$ denote the least common multiple of the orders of all elements of $G$. A $BH(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$. In this paper, we study the relation between $G$ and $h$ so that a $BH(G,h)$ matrix exists. We will only focus on $BH(\mathbb{Z}_n,h)$ matrices and $BH(G,2p^b)$ matrices, where $p$ is an odd prime. By our results, there are $2687$ open cases left for the existence of $BH(\mathbb{Z}_n,h)$ matrices in which $1\leq n,h \leq 100$. In the last section, we show that $BH(\mathbb{Z}_n,h)$ matrices can be used to construct a new family of perfect polyphase arrays.