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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