Some new orders of Hadamard and skew-Hadamard matrices

TL;DR

This paper presents novel constructions of Hadamard and skew-Hadamard matrices of specific orders using advanced combinatorial and algorithmic methods, expanding the known set of such matrices.

Contribution

It introduces new Hadamard and skew-Hadamard matrices of previously unconstructed orders using the Goethals-Seidel array and a novel hashing-based matching algorithm.

Findings

Constructed Hadamard matrices of orders 1004 and 2524.

Constructed skew-Hadamard matrices of orders 852 and 2524.

Developed an efficient matching algorithm based on hashing techniques.

Abstract

We construct Hadamard matrices of orders 4x251 = 1004 and 4x631 = 2524, and skew-Hadamard matrices of orders 4x213 = 852 and 4x631 = 2524. As far as we know, such matrices have not been constructed previously. The constructions use the Goethals-Seidel array, suitable supplementary difference sets on a cyclic group and a new efficient matching algorithm based on hashing techniques.