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

Abstract

An n by n skew-symmetric type . 1; 1/-matrix K D OEk i;j has 1's on the main diagonal and˙1's elsewhere with k i;j D k j;i . The largest possible determinant of such a matrix K is an interesting problem. The literature is extensive for n Á 0 mod 4 (skew-Hadamard matrices), but for n Á 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 D 11. To our knowledge, the upper bounds on the maximal determinant in orders 18 and 22 have been improved.

On Skew E–W Matrices

“…We recall that may attain the bound only when . For other orders, this question has been tackled in using the cocyclic approach.…”

“…We now make a sketch of a construction for skew E–W matrices. Skew E–W matrices have been found using this technique . This construction is made up of two steps.…”

“…Secondly, check if the matrices yielded by step 1 are equivalent to a skew matrix with 1's on the main diagonal. For this second step, we suggest the procedure given in , Section 3] where a backtracking search to decide whether or not a ( − 1, 1)‐matrix M of size n (given as input) is equivalent to a matrix of skew type is performed. If so, a skew‐matrix K equivalent to M is provided.…”

“…If so, a skew‐matrix K equivalent to M is provided. Naturally, the search tree is vast and methods to prune it are needed to go beyond the limits of the table produced in .…”

On quasi‐orthogonal cocycles

“…Recently, cocycles over groups G of even order not divisible by 4 have been examined as a source of ( − 1, 1)‐matrices with maximal determinant . In this paper, we discuss existence, classification, and combinatorics of such cocycles under the appropriate version of orthogonality—modifying a familiar balance condition on rows (and columns) of the cocyclic matrix when is divisible by 4.…”