CONSTRUCTIONS OF COMPLEX HADAMARD MATRICES VIA TILING ABELIAN GROUPS

Abstract. Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the con-nection between tiling of Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent very general construction of complex Hadamard matrices due to Dita [2] via a natural tiling construction. Then we ¯nd some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using an-other tiling construction, due to Szab¶o [8], 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 [10] of complex Hadamard matrices of small order