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

On quaternary complex Hadamard matrices of small orders

Ferenc Szöllősi 1,

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.
Keywords: Complex Hadamard matrix, fi ngerprint., Butson matrix.
Mathematics Subject Classification: Primary: 05B20; Secondary: 46L1.
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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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 Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[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  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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 Google Scholar

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