# New Constructions of Group-Invariant Butson Hadamard Matrices

## TL;DR

This paper introduces three new methods for constructing group-invariant Butson Hadamard matrices, including the first non-abelian case and two cases involving finite local rings, expanding the known classes of such matrices.

## Contribution

It presents the first known constructions of $ ext{BH}(G,h)$ matrices with non-abelian groups and introduces two new families based on finite local rings.

## Key findings

- First known family of $ ext{BH}(G,h)$ with non-abelian $G$
- Two new families of $ ext{BH}(G,h)$ with finite local rings
- Expands the existence and construction methods for group-invariant Butson Hadamard matrices

## Abstract

Let $G$ be a finite group and let $h$ be a positive integer. A $\text{BH}(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I_{|G|}$ is the identity matrix of order $|G|$. In this paper, we give three new constructions of $\text{BH}(G,h)$ matrices. The first construction is the first known family of $\text{BH}(G,h)$ matrices in which $G$ does not need to be abelian. The second and the third constructions are two families of $\text{BH}(G,h)$ matrices in which $G$ is a finite local ring.