Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 9, 2014

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

  • Víctor Álvarez EMAIL logo , José Andrés Armario , María Dolores Frau and Félix Gudiel
From the journal Open Mathematics


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.


[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, 858-87310.1016/j.laa.2011.05.018Search in Google Scholar

[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, 66-8210.13001/1081-3810.1580Search in Google Scholar

[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, 276-29010.1002/jcd.20191Search in Google Scholar

[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, 558-57010.1016/j.jsc.2007.06.009Search in Google Scholar

[5] Armario J.A., Frau M.D., Self-dual codes from (1; 1)-matrices of skew type, preprint available at in Google Scholar

[6] Bussemaker F., Kaplansky I., McKay B., Seidel J., Determinants of matrices of the conference type, Linear Algebra Appl., 1997, 261, 275-29210.1016/S0024-3795(96)00412-0Search in Google Scholar

[7] Cameron P., Problem 104 (Peter Cameron’s Blog), <> 2011, accessed 26 September 2013Search in Google Scholar

[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, 161-171Search in Google Scholar

[9] Ehlich H., Determiantenabschätzungen für binäre Matrizen, Math. Z., 1964, 83, 123-13210.1007/BF01111249Search in Google Scholar

[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, 252-25310.1016/j.disc.2004.05.009Search in Google Scholar

[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), 129Search in Google Scholar

[12] Horadam K.J., Hadamard Matrices and Their Applications, Princeton University Press, Princeton, NJ, 200710.1515/9781400842902Search in Google Scholar

[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, 2006Search in Google Scholar

[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, 2006Search in Google Scholar

[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, 2054-206610.1016/j.laa.2011.10.030Search in Google Scholar

[16] MacLane S., Homology, Classics in Mathematics Springer-Verlang, Berlin, 1995, Reprint of the 1975 editionSearch in Google Scholar

[17] Orrick W., Solomon B., The Hadamard Maximal Determinant Problem (website),, accessed 3 October 2013Search in Google Scholar

[18] Szollosi F., Exotic complex Hadamard matrices and their equivalence, Cryptogr. Commun., 2010, 2, 187-19810.1007/s12095-010-0021-3Search in Google Scholar

[19] Wojtas W., On Hadamard’s inequallity for the determinants of order non-divisible by 4, Colloq. Math., 1964, 12, 73-83 10.4064/cm-12-1-73-83Search in Google Scholar

Received: 2013-12-4
Accepted: 2014-4-15
Published Online: 2014-10-9
Published in Print: 2015-1-1

© 2015 Víctor Álvarez et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 27.2.2024 from
Scroll to top button