Constructions of complex Hadamard matrices via tiling Abelian groups

TL;DR

This paper explores how tiling Abelian groups can be used to construct and analyze complex Hadamard matrices, introducing new parametric families and conditions for their equivalence to known types.

Contribution

It links tiling group theory with complex Hadamard matrix construction, recovering Dita's method, establishing necessary conditions, and discovering new matrix families.

Findings

Recovered Dita's construction via tiling groups

Established necessary conditions for Dita-type equivalence

Discovered new parametric families of complex Hadamard matrices

Abstract

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent very general construction of complex Hadamard matrices due to Dita via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabo, we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue of complex Hadamard matrices of small order.