May  2011, 5(2): 309-315. doi: 10.3934/amc.2011.5.309

On quaternary complex Hadamard matrices of small orders

1. 

Department of Mathematics and its Applications, Central European University, H-1051, Nádor u. 9, Budapest, Hungary

Received  May 2010 Revised  April 2011 Published  May 2011

One of the main goals of design theory is to classify, characterize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete enumeration of the objects is possible, there is no apparent way how to study them in details, store them efficiently, or generate a particular one rapidly. In this paper we propose a novel method to deal with these difficulties, and illustrate it by presenting the classification of quaternary complex Hadamard matrices up to order $8$. The obtained matrices are members of only a handful of parametric families, and each inequivalent matrix, up to transposition, can be identified through its fingerprint.
Citation: Ferenc Szöllősi. On quaternary complex Hadamard matrices of small orders. Advances in Mathematics of Communications, 2011, 5 (2) : 309-315. doi: 10.3934/amc.2011.5.309
References:
[1]

G. Auberson, A. Martin and G. Mennessier, On the reconstruction of a unitary matrix from its moduli, Commun. Math. Phys., 140 (1991), 417-436. doi: 10.1007/BF02099133.

[2]

S. Bouguezel, M. O. Ahmed and M. N. S. Swami, A new class of reciprocal-orthogonal parametric transforms, IEEE Trans. Circuits Syst. I, 56 (2009), 795-805. doi: 10.1109/TCSI.2008.2002923.

[3]

A. T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc., 13 (1962), 894-898. doi: 10.1090/S0002-9939-1962-0142557-0.

[4]

R. Craigen, Equivalence classes of inverse orthogonal and unit Hadamard matrices, Bull. Austral. Math. Soc., 44 (1991), 109-115. doi: 10.1017/S0004972700029506.

[5]

P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A, 20 (2004), 5355-5374.

[6]

P. Diţă, Complex Hadamard matrices from Sylvester inverse orthogonal matrices, Open Sys. Inform. Dyn., 16 (2009), 387-405; see the errata at arXiv:0901.0982v2

[7]

P. Diţă, Hadamard Matrices from mutually unbiased bases, J. Math. Phys., 51 (2010), 20.

[8]

T. Durt, B.-H. Englert, I. Bengtsson and K. Życzkowski, On mutually unbiased bases, Intern. J. Quantum Inform., 8 (2010), 535-640. doi: 10.1142/S0219749910006502.

[9]

U. Haagerup, Orthogonal maximal Abelian *-subalgebras of $n\times n$ matrices and cyclic $n$-roots, in "Operator Algebras and Quantum Field Theory (Rome),'' MA International Press, (1996), 296-322.

[10]

M. Harada, C. Lam and V. D. Tonchev, Symmetric $(4,4)$-nets and generalized Hadamard matrices over groups of order $4$, Des. Codes Crypt., 34 (2005), 71-87. doi: 10.1007/s10623-003-4195-y.

[11]

K. J. Horadam, "Hadamard Matrices and Their Applications,'' Princeton University Press, Princeton, 2007.

[12]

M. N. Kolountzakis and M. Matolcsi, Complex Hadamard matrices and the spectral set conjecture, Collectanea Math., Extra (2006), 281-291.

[13]

M. H. Lee and V. V. Vavrek, Jacket conference matrices and Paley transformation, in "Eleventh International Wokshop on Algebraic and Combinatorial Coding Theory,'' Pamporovo, Bulgaria, (2008), 181-185.

[14]

M. Matolcsi, J. Réffy and F. Szöllősi, Constructions of complex Hadamard matrices via tiling abelian groups, Open Sys. Inform. Dyn., 14 (2007), 247-263. doi: 10.1007/s11080-007-9050-6.

[15]

T. S. Michael and W. D. Wallis, Skew-Hadamard matrices and the Smith normal form, Des. Codes Crypt., 13 (1998), 173-176. doi: 10.1023/A:1008230429804.

[16]

S. Popa, Orthogonal pairs of *-subalgebras in finite von Neumann algebras, J. Operator Theory, 9 (1983), 253-268.

[17]

F. Szöllősi, Parametrizing complex Hadamard matrices, European J. Combin., 29 (2008), 1219-1234. doi: 10.1016/j.ejc.2007.06.009.

[18]

F. Szöllősi, Exotic complex Hadamard matrices and their equivalence, Crypt. Commun., 2 (2010), 187-198. doi: 10.1007/s12095-010-0021-3.

[19]

W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inform. Dyn., 13 (2006), 133-177. doi: 10.1007/s11080-006-8220-2.

[20]

T. Tao, Fuglede's conjecture is false in $5$ and higher dimensions, Math Res. Letters, 11 (2004), 251-258.

[21]

R. F. Werner, All teleportation and dense coding schemes, J. Phys. A, 34 (2001), 7081-7094. doi: 10.1088/0305-4470/34/35/332.

[22]

G. Zauner, "Quantendesigns: Grundzäuge einer nichtkommutativen Designtheorie'' (in German), Ph.D thesis, Universität Wien, 1999; available online at http://www.mat.univie.ac.at/~neum/ms/zauner.pdf

show all references

References:
[1]

G. Auberson, A. Martin and G. Mennessier, On the reconstruction of a unitary matrix from its moduli, Commun. Math. Phys., 140 (1991), 417-436. doi: 10.1007/BF02099133.

[2]

S. Bouguezel, M. O. Ahmed and M. N. S. Swami, A new class of reciprocal-orthogonal parametric transforms, IEEE Trans. Circuits Syst. I, 56 (2009), 795-805. doi: 10.1109/TCSI.2008.2002923.

[3]

A. T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc., 13 (1962), 894-898. doi: 10.1090/S0002-9939-1962-0142557-0.

[4]

R. Craigen, Equivalence classes of inverse orthogonal and unit Hadamard matrices, Bull. Austral. Math. Soc., 44 (1991), 109-115. doi: 10.1017/S0004972700029506.

[5]

P. Diţă, Some results on the parametrization of complex Hadamard matrices, J. Phys. A, 20 (2004), 5355-5374.

[6]

P. Diţă, Complex Hadamard matrices from Sylvester inverse orthogonal matrices, Open Sys. Inform. Dyn., 16 (2009), 387-405; see the errata at arXiv:0901.0982v2

[7]

P. Diţă, Hadamard Matrices from mutually unbiased bases, J. Math. Phys., 51 (2010), 20.

[8]

T. Durt, B.-H. Englert, I. Bengtsson and K. Życzkowski, On mutually unbiased bases, Intern. J. Quantum Inform., 8 (2010), 535-640. doi: 10.1142/S0219749910006502.

[9]

U. Haagerup, Orthogonal maximal Abelian *-subalgebras of $n\times n$ matrices and cyclic $n$-roots, in "Operator Algebras and Quantum Field Theory (Rome),'' MA International Press, (1996), 296-322.

[10]

M. Harada, C. Lam and V. D. Tonchev, Symmetric $(4,4)$-nets and generalized Hadamard matrices over groups of order $4$, Des. Codes Crypt., 34 (2005), 71-87. doi: 10.1007/s10623-003-4195-y.

[11]

K. J. Horadam, "Hadamard Matrices and Their Applications,'' Princeton University Press, Princeton, 2007.

[12]

M. N. Kolountzakis and M. Matolcsi, Complex Hadamard matrices and the spectral set conjecture, Collectanea Math., Extra (2006), 281-291.

[13]

M. H. Lee and V. V. Vavrek, Jacket conference matrices and Paley transformation, in "Eleventh International Wokshop on Algebraic and Combinatorial Coding Theory,'' Pamporovo, Bulgaria, (2008), 181-185.

[14]

M. Matolcsi, J. Réffy and F. Szöllősi, Constructions of complex Hadamard matrices via tiling abelian groups, Open Sys. Inform. Dyn., 14 (2007), 247-263. doi: 10.1007/s11080-007-9050-6.

[15]

T. S. Michael and W. D. Wallis, Skew-Hadamard matrices and the Smith normal form, Des. Codes Crypt., 13 (1998), 173-176. doi: 10.1023/A:1008230429804.

[16]

S. Popa, Orthogonal pairs of *-subalgebras in finite von Neumann algebras, J. Operator Theory, 9 (1983), 253-268.

[17]

F. Szöllősi, Parametrizing complex Hadamard matrices, European J. Combin., 29 (2008), 1219-1234. doi: 10.1016/j.ejc.2007.06.009.

[18]

F. Szöllősi, Exotic complex Hadamard matrices and their equivalence, Crypt. Commun., 2 (2010), 187-198. doi: 10.1007/s12095-010-0021-3.

[19]

W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inform. Dyn., 13 (2006), 133-177. doi: 10.1007/s11080-006-8220-2.

[20]

T. Tao, Fuglede's conjecture is false in $5$ and higher dimensions, Math Res. Letters, 11 (2004), 251-258.

[21]

R. F. Werner, All teleportation and dense coding schemes, J. Phys. A, 34 (2001), 7081-7094. doi: 10.1088/0305-4470/34/35/332.

[22]

G. Zauner, "Quantendesigns: Grundzäuge einer nichtkommutativen Designtheorie'' (in German), Ph.D thesis, Universität Wien, 1999; available online at http://www.mat.univie.ac.at/~neum/ms/zauner.pdf

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