Complex Hadamard Matrices contained in a Bose-Mesner algebra

Takuya Ikuta, Akihiro Munemasa

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


Acomplex 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 Cd(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.

Original languageEnglish
Pages (from-to)91-110
Number of pages20
JournalSpecial Matrices
Issue number1
Publication statusPublished - 2015 Jan


  • Association scheme
  • Complex Hadamard matrix
  • Type-II matrix

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology


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