Complex Hadamard Matrices contained in a Bose–Mesner algebra

Takuya Ikuta; Akihiro Munemasa

Special Matrices (2015)

  • Volume: 3, Issue: 1, page 91-110, electronic only
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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Takuya Ikuta, and Akihiro Munemasa. "Complex Hadamard Matrices contained in a Bose–Mesner algebra." Special Matrices 3.1 (2015): 91-110, electronic only. <http://eudml.org/doc/270994>.

@article{TakuyaIkuta2015,
abstract = {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.},
author = {Takuya Ikuta, Akihiro Munemasa},
journal = {Special Matrices},
keywords = {association scheme; complex Hadamard matrix; type-II matrix},
language = {eng},
number = {1},
pages = {91-110, electronic only},
title = {Complex Hadamard Matrices contained in a Bose–Mesner algebra},
url = {http://eudml.org/doc/270994},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Takuya Ikuta
AU - Akihiro Munemasa
TI - Complex Hadamard Matrices contained in a Bose–Mesner algebra
JO - Special Matrices
PY - 2015
VL - 3
IS - 1
SP - 91
EP - 110, electronic only
AB - 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.
LA - eng
KW - association scheme; complex Hadamard matrix; type-II matrix
UR - http://eudml.org/doc/270994
ER -

References

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  11. [11] R. Hosoya and H. Suzuki, Type II matrices and their Bose-Mesner algebras, J. Algebraic Combin. 17 (2003), 19–37. Zbl1011.05065
  12. [12] F. Jaeger, M. Matsumoto, and K. Nomura, Bose-Mesner algebras related to type II matrices and spin models, J. Algebraic Combin. 8 (1998), 39–72. Zbl0974.05084
  13. [13] R. Nicoara, A finiteness result for commuting squares of matrix algebras, J. Operator Theory 55 (2006), 101–116. Zbl1115.46052
  14. [14] K. Nomura, Type II matrices of size five, Graphs Combin.15 (1999), 79–92. Zbl0935.05022
  15. [15] A. D. Sankey, Type-II matrices in weighted Bose-Mesner algebras of ranks 2 and 3, J. Algebraic Combin. 32 (2010), 133–153. Zbl1230.05298
  16. [16] F. Szöllősi, Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun. 2 (2010), no. 2, 187–198. Zbl1228.05097
  17. [17] W. Tadej and K. Życzkowski A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006), 133–177.[Crossref] 

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