American Institute of Mathematical Sciences

Advanced
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]

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

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

[1]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
[2]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247
[3]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2559-2599. doi: 10.3934/dcds.2020375
[4]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406
[5]

Xiaoyi Zhou, Tong Ye, Tony T. Lee. Designing and analysis of a Wi-Fi data offloading strategy catering for the preference of mobile users. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021038
[6]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377
[7]

Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021047
[8]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056
[9]

Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021061
[10]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[11]

Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021005

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences