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

Víctor Álvarez; José Andrés Armario; María Dolores Frau; Félix Gudiel

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 16-25, electronic only
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Víctor Álvarez, et al. "Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach." Open Mathematics 13.1 (2015): 16-25, electronic only. <http://eudml.org/doc/268889>.

@article{VíctorÁlvarez2015,
abstract = {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.},
author = {Víctor Álvarez, José Andrés Armario, María Dolores Frau, Félix Gudiel},
journal = {Open Mathematics},
keywords = {(-1; 1)-matrix of skew type; Cocyclic matrices; Maximal determinants; -matrix of skew type; cocyclic matrices; maximal determinants},
language = {eng},
number = {1},
pages = {16-25, electronic only},
title = {Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach},
url = {http://eudml.org/doc/268889},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Víctor Álvarez
AU - José Andrés Armario
AU - María Dolores Frau
AU - Félix Gudiel
TI - Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 16
EP - 25, electronic only
AB - 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.
LA - eng
KW - (-1; 1)-matrix of skew type; Cocyclic matrices; Maximal determinants; -matrix of skew type; cocyclic matrices; maximal determinants
UR - http://eudml.org/doc/268889
ER -

References

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  1. [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] Zbl1236.05043
  2. [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 Zbl1253.05050
  3. [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] Zbl1182.05019
  4. [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] Zbl1163.05004
  5. [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. [6] Bussemaker F., Kaplansky I., McKay B., Seidel J., Determinants of matrices of the conference type, Linear Algebra Appl., 1997, 261, 275-292 Zbl0878.15001
  7. [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. [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 Zbl0735.05017
  9. [9] Ehlich H., Determiantenabschätzungen für binäre Matrizen, Math. Z., 1964, 83, 123-132 
  10. [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 Zbl1054.62081
  11. [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 Zbl0785.05019
  12. [12] Horadam K.J., Hadamard Matrices and Their Applications, Princeton University Press, Princeton, NJ, 2007 Zbl1145.05014
  13. [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. [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. [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] Zbl1236.15017
  16. [16] MacLane S., Homology, Classics in Mathematics Springer-Verlang, Berlin, 1995, Reprint of the 1975 edition 
  17. [17] Orrick W., Solomon B., The Hadamard Maximal Determinant Problem (website), http://www.indiana.edu/~maxdet/, accessed 3 October 2013 
  18. [18] Szollosi F., Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun., 2010, 2, 187-198 Zbl1228.05097
  19. [19] Wojtas W., On Hadamard’s inequallity for the determinants of order non-divisible by 4, Colloq. Math., 1964, 12, 73-83 Zbl0126.02604

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