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
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show all references

References:
[1]

Commun. Math. Phys., 140 (1991), 417-436. doi: 10.1007/BF02099133.  Google Scholar

[2]

IEEE Trans. Circuits Syst. I, 56 (2009), 795-805. doi: 10.1109/TCSI.2008.2002923.  Google Scholar

[3]

Proc. Amer. Math. Soc., 13 (1962), 894-898. doi: 10.1090/S0002-9939-1962-0142557-0.  Google Scholar

[4]

Bull. Austral. Math. Soc., 44 (1991), 109-115. doi: 10.1017/S0004972700029506.  Google Scholar

[5]

J. Phys. A, 20 (2004), 5355-5374.  Google Scholar

[6]

Open Sys. Inform. Dyn., 16 (2009), 387-405; see the errata at arXiv:0901.0982v2  Google Scholar

[7]

J. Math. Phys., 51 (2010), 20.  Google Scholar

[8]

Intern. J. Quantum Inform., 8 (2010), 535-640. doi: 10.1142/S0219749910006502.  Google Scholar

[9]

in "Operator Algebras and Quantum Field Theory (Rome),'' MA International Press, (1996), 296-322.  Google Scholar

[10]

Des. Codes Crypt., 34 (2005), 71-87. doi: 10.1007/s10623-003-4195-y.  Google Scholar

[11]

Princeton University Press, Princeton, 2007.  Google Scholar

[12]

Collectanea Math., Extra (2006), 281-291.  Google Scholar

[13]

in "Eleventh International Wokshop on Algebraic and Combinatorial Coding Theory,'' Pamporovo, Bulgaria, (2008), 181-185. Google Scholar

[14]

Open Sys. Inform. Dyn., 14 (2007), 247-263. doi: 10.1007/s11080-007-9050-6.  Google Scholar

[15]

Des. Codes Crypt., 13 (1998), 173-176. doi: 10.1023/A:1008230429804.  Google Scholar

[16]

J. Operator Theory, 9 (1983), 253-268.  Google Scholar

[17]

European J. Combin., 29 (2008), 1219-1234. doi: 10.1016/j.ejc.2007.06.009.  Google Scholar

[18]

Crypt. Commun., 2 (2010), 187-198. doi: 10.1007/s12095-010-0021-3.  Google Scholar

[19]

Open Syst. Inform. Dyn., 13 (2006), 133-177. doi: 10.1007/s11080-006-8220-2.  Google Scholar

[20]

Math Res. Letters, 11 (2004), 251-258.  Google Scholar

[21]

J. Phys. A, 34 (2001), 7081-7094. doi: 10.1088/0305-4470/34/35/332.  Google Scholar

[22]

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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