On quaternary complex Hadamard matrices of small orders

TL;DR

This paper introduces a new method for classifying small-order quaternary complex Hadamard matrices, efficiently identifying inequivalent matrices within parametric families using a fingerprinting technique.

Contribution

It presents a novel approach to classify and distinguish quaternary complex Hadamard matrices up to order 8, addressing enumeration and storage challenges.

Findings

Classified all quaternary complex Hadamard matrices up to order 8.

Identified matrices within a few parametric families.

Developed a fingerprinting method for matrix identification.

Abstract

One of the main goals of design theory is to classify, characterize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete enumeration of the objects is possible, there is no apparent way how to study them in details, store them efficiently, or generate a particular one rapidly. In this paper we propose a novel method to deal with these difficulties, and illustrate it by presenting the classification of quaternary complex Hadamard matrices up to order 8. The obtained matrices are members of only a handful of parametric families, and each inequivalent matrix, up to transposition, can be identified through its fingerprint.