# 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

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topTakuya 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

top- [1] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984. Zbl0555.05019
- [2] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235–265. Zbl0898.68039
- [3] A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989. Zbl0747.05073
- [4] A. Chan, Complex Hadamard matrices and strongly regular graphs, arXiv:1102.5601.
- [5] A. Chan and C. Godsil, Type-II matrices and combinatorial structures, Combinatorica, 30 (2010), 1–24. [WoS][Crossref] Zbl1224.05502
- [6] A. Chan and R. Hosoya, Type-II matrices attached to conference graphs, J. Algebraic Combin. 20 (2004), 341–351. Zbl1056.05148
- [7] R. Craigen, Equivalence classes of inverse orthogonal and unit Hadamard matrices, Bull. Austral. Math. Soc. 44 (1991), no. 1, 109–115. [Crossref] Zbl0719.05016
- [8] E. van Dam, Three-class association schemes, J. Algebraic Combin. 10 (1999), 69–107. Zbl0929.05096
- [9] J. M. Goethals and J. J. Seidel, Strongly regular graphs derived from combinatorial designs, Can. J.Math., 22, (1970), 597–614. Zbl0198.29301
- [10] U. Haagerup, Orthogonal maximal Abelian *-subalgebras of n × n matrices and cyclic n-roots, Operator Algebras and Quantum Field Theory (Rome), Cambridge, MA, International Press, (1996), 296–322.
- [11] R. Hosoya and H. Suzuki, Type II matrices and their Bose-Mesner algebras, J. Algebraic Combin. 17 (2003), 19–37. Zbl1011.05065
- [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] R. Nicoara, A finiteness result for commuting squares of matrix algebras, J. Operator Theory 55 (2006), 101–116. Zbl1115.46052
- [14] K. Nomura, Type II matrices of size five, Graphs Combin.15 (1999), 79–92. Zbl0935.05022
- [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] F. Szöllősi, Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun. 2 (2010), no. 2, 187–198. Zbl1228.05097
- [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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