Equivalence of the Existence of Hadamard Matrices and Cretan$(4t-1,2)$-Mersenne Matrices

TL;DR

This paper establishes an equivalence between the existence of Hadamard matrices of order 4t and Cretan(4t-1,2)-Mersenne matrices, linking two important classes of orthogonal matrices.

Contribution

It proves for the first time that the existence of Hadamard matrices of order 4t is equivalent to the existence of Cretan(4t-1,2)-Mersenne matrices.

Findings

Equivalence between Hadamard and Cretan Mersenne matrices established

Provides a new perspective on the existence problem of orthogonal matrices

Bridges two classes of matrices in combinatorial design theory

Abstract

We study orthogonal matrices whose elements have moduli $\leq 1$. This paper shows that the existence of two such families of matrices is equivalent. Specifically we show that the existence of an Hadamard matrix of order $4t$ is equivalent to the existence of a Cretan$(4t-1,2)$-Mersenne matrix.