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arXiv · 1411.0057

Complex Hadamard matrices contained in a Bose-Mesner algebra

Takuya Ikuta·Akihiro Munemasa
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Abstract

A complex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying $HH^*= nI$, where $*$ stands for the Hermitian transpose and I is the identity matrix of order $n$. In this paper, we first determine the image of a certain rational map from the $d$-dimensional complex projective space to $\mathbb{C}^{d(d+1)/2}$. Applying this result with $d=3$, we give constructions of complex Hadamard matrices, and more generally, type-II matrices, in the Bose-Mesner algebra of a certain 3-class symmetric association scheme. In particular, we recover the complex Hadamard matrices of order 15 found by Ada Chan. We compute the Haagerup sets to show inequivalence of resulting type-II matrices, and determine the Nomura algebras to show that the resulting matrices are not decomposable into generalized tensor products.

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Takuya Ikuta, Akihiro Munemasa. 2014-11-01. Complex Hadamard matrices contained in a Bose-Mesner algebra. https://doi.org/10.1515/spma-2015-0009

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