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Applications in quantum information theory and quantum tomography have\udraised current interest in complex Hadamard matrices. In this note we investigate the connection\udbetween tiling of Abelian groups and constructions of complex Hadamard matrices.\udFirst, we recover a recent very general construction of complex Hadamard matrices due to\udDita [2] via a natural tiling construction. Then we find some necessary conditions for any\udgiven complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another\udtiling construction, due to Szab´o [8], we arrive at new parametric families of complex\udHadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove\udthat these families do not arise with Dita’s construction. These new families complement\udthe recent catalogue [10] of complex Hadamard matrices of small order

Constructions of Complex Hadamard Matrices via Tiling Abelian Groups