Orderly generation of Butson Hadamard matrices

TL;DR

This paper classifies small-order Butson-type complex Hadamard matrices using computer-aided methods, providing enumeration, new examples, and nonexistence results for various parameters.

Contribution

It offers the first enumeration of certain small-order matrices, introduces a new example, and establishes nonexistence results for specific parameters.

Findings

Enumerated $ ext{BH}(21,3)$, $ ext{BH}(16,4)$, and $ ext{BH}(14,6)$ matrices.

Presented the first known $ ext{BH}(14,10)$ matrix.

Proved nonexistence of certain matrices such as $ ext{BH}(8,15)$ and others.

Abstract

In this paper Butson-type complex Hadamard matrices $\mathrm{BH}(n,q)$ of order $n$ and complexity $q$ are classified for small parameters by computer-aided methods. Our main results include the enumeration of $\mathrm{BH}(21,3)$, $\mathrm{BH}(16,4)$, and $\mathrm{BH}(14,6)$ matrices. There are exactly $72$, 1786763, and $167776$ such matrices, up to monomial equivalence. Additionally, we show an example of a $\mathrm{BH}(14,10)$ matrix for the first time, and show the nonexistence of $\mathrm{BH}(8,15)$, $\mathrm{BH}(11,q)$ for $q\in\{10,12,14,15\}$, and $\mathrm{BH}(13,10)$ matrices.