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.  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.  doi: 10.1109/TCSI.2008.2002923.  Google Scholar

[3]

A. T. Butson, Generalized Hadamard matrices,, Proc. Amer. Math. Soc., 13 (1962), 894.  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.  doi: 10.1017/S0004972700029506.  Google Scholar

[5]

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

[6]

P. Diţă, Complex Hadamard matrices from Sylvester inverse orthogonal matrices,, Open Sys. Inform. Dyn., 16 (2009), 387.   Google Scholar

[7]

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

[8]

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

[9]

U. Haagerup, Orthogonal maximal Abelian *-subalgebras of $n\times n$ matrices and cyclic $n$-roots,, in, (1996), 296.   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.  doi: 10.1007/s10623-003-4195-y.  Google Scholar

[11]

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

[12]

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

[13]

M. H. Lee and V. V. Vavrek, Jacket conference matrices and Paley transformation,, in, (2008), 181.   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.  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.  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.   Google Scholar

[17]

F. Szöllősi, Parametrizing complex Hadamard matrices,, European J. Combin., 29 (2008), 1219.  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.  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.  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.   Google Scholar

[21]

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

[22]

G. Zauner, "Quantendesigns: Grundzäuge einer nichtkommutativen Designtheorie'' (in German),, Ph.D thesis, (1999).   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.  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.  doi: 10.1109/TCSI.2008.2002923.  Google Scholar

[3]

A. T. Butson, Generalized Hadamard matrices,, Proc. Amer. Math. Soc., 13 (1962), 894.  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.  doi: 10.1017/S0004972700029506.  Google Scholar

[5]

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

[6]

P. Diţă, Complex Hadamard matrices from Sylvester inverse orthogonal matrices,, Open Sys. Inform. Dyn., 16 (2009), 387.   Google Scholar

[7]

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

[8]

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

[9]

U. Haagerup, Orthogonal maximal Abelian *-subalgebras of $n\times n$ matrices and cyclic $n$-roots,, in, (1996), 296.   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.  doi: 10.1007/s10623-003-4195-y.  Google Scholar

[11]

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

[12]

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

[13]

M. H. Lee and V. V. Vavrek, Jacket conference matrices and Paley transformation,, in, (2008), 181.   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.  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.  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.   Google Scholar

[17]

F. Szöllősi, Parametrizing complex Hadamard matrices,, European J. Combin., 29 (2008), 1219.  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.  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.  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.   Google Scholar

[21]

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

[22]

G. Zauner, "Quantendesigns: Grundzäuge einer nichtkommutativen Designtheorie'' (in German),, Ph.D thesis, (1999).   Google Scholar

[1]

Francis C. Motta, Patrick D. Shipman. Informing the structure of complex Hadamard matrix spaces using a flow. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2349-2364. doi: 10.3934/dcdss.2019147

[2]

Giuseppe Geymonat, Françoise Krasucki. Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains. Communications on Pure & Applied Analysis, 2009, 8 (1) : 295-309. doi: 10.3934/cpaa.2009.8.295

[3]

Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014

[4]

Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing, 2019, 2 (2) : 107-126. doi: 10.3934/mfc.2019009

[5]

Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009

[6]

Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015

[7]

K. T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026

[8]

Lei Zhang, Anfu Zhu, Aiguo Wu, Lingling Lv. Parametric solutions to the regulator-conjugate matrix equations. Journal of Industrial & Management Optimization, 2017, 13 (2) : 623-631. doi: 10.3934/jimo.2016036

[9]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55

[10]

Houduo Qi, ZHonghang Xia, Guangming Xing. An application of the nearest correlation matrix on web document classification. Journal of Industrial & Management Optimization, 2007, 3 (4) : 701-713. doi: 10.3934/jimo.2007.3.701

[11]

Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75

[12]

A. Cibotarica, Jiu Ding, J. Kolibal, Noah H. Rhee. Solutions of the Yang-Baxter matrix equation for an idempotent. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 347-352. doi: 10.3934/naco.2013.3.347

[13]

Haixia Liu, Jian-Feng Cai, Yang Wang. Subspace clustering by (k,k)-sparse matrix factorization. Inverse Problems & Imaging, 2017, 11 (3) : 539-551. doi: 10.3934/ipi.2017025

[14]

Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks & Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63

[15]

Debasisha Mishra. Matrix group monotonicity using a dominance notion. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 267-274. doi: 10.3934/naco.2015.5.267

[16]

Lingling Lv, Zhe Zhang, Lei Zhang, Weishu Wang. An iterative algorithm for periodic sylvester matrix equations. Journal of Industrial & Management Optimization, 2018, 14 (1) : 413-425. doi: 10.3934/jimo.2017053

[17]

Joshua Du, Jun Ji. An integral representation of the determinant of a matrix and its applications. Conference Publications, 2005, 2005 (Special) : 225-232. doi: 10.3934/proc.2005.2005.225

[18]

Boris Baeumer, Lipika Chatterjee, Peter Hinow, Thomas Rades, Ami Radunskaya, Ian Tucker. Predicting the drug release kinetics of matrix tablets. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 261-277. doi: 10.3934/dcdsb.2009.12.261

[19]

A. Chauviere, L. Preziosi, T. Hillen. Modeling the motion of a cell population in the extracellular matrix. Conference Publications, 2007, 2007 (Special) : 250-259. doi: 10.3934/proc.2007.2007.250

[20]

Yi Yang, Jianwei Ma, Stanley Osher. Seismic data reconstruction via matrix completion. Inverse Problems & Imaging, 2013, 7 (4) : 1379-1392. doi: 10.3934/ipi.2013.7.1379

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]