A nonexistence criterion and new constructions for Butson Hadamard matrices

TL;DR

This paper introduces a new necessary condition for the existence of Butson Hadamard matrices, leading to nonexistence results, computational discoveries, and new constructions of matrices with specific parameters.

Contribution

It presents a novel nonexistence criterion based on positive definite functions and constructs new matrices using cyclotomic cosets, expanding known examples of Butson Hadamard matrices.

Findings

Proved nonexistence for certain (n,q) parameters.

Discovered a new BH(18,14) matrix via computer search.

Constructed new BH matrices with 2-circulant structure for specific parameters.

Abstract

Based on the concept of positive definite functions on finite groups, we present a new necessary condition for the existence of Butson Hadamard matrices $BH(n,q)$. We use this condition to prove some nonexistence results for a sequence of values of $(n,q)$, and also to facilitate a computer search and discover a matrix $BH(18,14)$. Furthermore, we use cyclotomic cosets to construct matrices $BH(34,10)$, $BH(62,6)$, $BH(82,6)$, and $BH(146,6)$ for the first time. These matrices have a $2$-circulant structure.