Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach

• Autorzy: Víctor Álvarez , José Andrés Armario , María Dolores Frau , Félix Gudiel
• Języki publikacji: EN

Abstrakty

EN

An n by n skew-symmetric type (-1; 1)-matrix K =[ki;j ] has 1’s on the main diagonal and ±1’s elsewhere with ki;j =-kj;i . The largest possible determinant of such a matrix K is an interesting problem. The literature is extensive for n ≡ 0 mod 4 (skew-Hadamard matrices), but for n ≡ 2 mod 4 there are few results known for this question. In this paper we approach this problem constructing cocyclic matrices over the dihedral group of 2t elements, for t odd, which are equivalent to (-1; 1)-matrices of skew type. Some explicit calculations have been done up to t =11. To our knowledge, the upper bounds on the maximal determinant in orders 18 and 22 have been improved.

Słowa kluczowe

EN

(-1; 1)-matrix of skew type; Cocyclic matrices; Maximal determinants

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

wydano

2015-01-01

otrzymano

2013-12-04

zaakceptowano

2014-04-15

online

2014-10-09

Twórcy

autor

Víctor Álvarez

• Departamento de Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012, Sevilla, Spain

autor

José Andrés Armario

• Departamento de Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012, Sevilla, Spain

autor

María Dolores Frau

• Departamento de Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012, Sevilla, Spain

autor

Félix Gudiel

• Departamento de Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012, Sevilla, Spain

Bibliografia

• [1] Álvarez V., Armario J.A., Frau M.D., Gudiel F., The maximal determinant of cocyclic (1; 1)-matrices over D2t , Linear Algebra Appl., 2012, 436, 858-873[WoS]
• [2] Álvarez V., Armario J.A., Frau M.D., Gudiel F., Embedding cocyclic D-optimal designs in cocyclic Hadamard matrices, Electron. J. Linear Algebra, 2012, 24, 66-82
• [3] Álvarez V., Armario J.A., Frau M.D., Real P., A system of equations for describing cocyclic Hadamard matrices, J. Comb. Des., 2008, 16, 276-290[WoS][Crossref]
• [4] Álvarez V., Armario J.A., Frau M.D., Real P., The homological reduction method for computing cocyclic Hadamard matrices, J. Symb. Comput., 2009, 44, 558-570[WoS]
• [5] Armario J.A., Frau M.D., Self-dual codes from (1; 1)-matrices of skew type, preprint available at http://arxiv.org/abs/1311.2637
• [6] Bussemaker F., Kaplansky I., McKay B., Seidel J., Determinants of matrices of the conference type, Linear Algebra Appl., 1997, 261, 275-292
• [7] Cameron P., Problem 104 (Peter Cameron’s Blog), <http://cameroncounts.wordpress.com/2011/08/19/a-matrix-problem/> 2011, accessed 26 September 2013
• [8] Craigen R., The range of the determinant function on the set of n x n .0; 1)-matrices, J. Combin. Math. Combin. Comput. 1990, 8, 161-171
• [9] Ehlich H., Determiantenabschätzungen für binäre Matrizen, Math. Z., 1964, 83, 123-132
• [10] Fletcher R. J., Koukouvinos C., Seberry J., New skew-Hadamard matrices of order 4 · 59 and new D-optimal designs of order 2 · 59, Discrete Math., 2004, 286, 252-253
• [11] Horadam K.J., de Launey W., Cocyclic development of designs, J. Algebraic Combin. 2 (3) (1993) 267-290; Erratum: J. Algebraic Combin. 1994, 3 (1), 129
• [12] Horadam K.J., Hadamard Matrices and Their Applications, Princeton University Press, Princeton, NJ, 2007
• [13] Ionin Y., Kharaghani H., Balanced generalized Weighing matrices and Conference matrices, in: C. Colbourn and J. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, 2nd ed., Taylor and Francis, Boca Raton, 2006
• [14] Kharaghani H., Orrick W., D-optimal designs, in: C. Colbourn and J. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, 2nd ed., Taylor and Francis, Boca Raton, 2006
• [15] Krapiperi A., Mitrouli M., Neubauer M.G., Seberry J., An eigenvalue approach evaluating minors for weighing matrices W(n, n - 1), Linear Algebra Appl., 2012, 436, 2054-2066[WoS]
• [16] MacLane S., Homology, Classics in Mathematics Springer-Verlang, Berlin, 1995, Reprint of the 1975 edition
• [17] Orrick W., Solomon B., The Hadamard Maximal Determinant Problem (website), http://www.indiana.edu/~maxdet/, accessed 3 October 2013
• [18] Szollosi F., Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun., 2010, 2, 187-198
• [19] Wojtas W., On Hadamard’s inequallity for the determinants of order non-divisible by 4, Colloq. Math., 1964, 12, 73-83

Identyfikatory

DOI

10.1515/math-2015-0003