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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

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.

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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.

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