Cretan(4t+1) Matrices

TL;DR

This paper introduces a method to construct infinitely many new Cretan(4t+1) matrices of various orders using Hadamard, weighing, and generalized Hadamard matrices, expanding known examples significantly.

Contribution

It provides a new construction framework for Cretan(4t+1) matrices for all orders n ≥ 3, utilizing advanced combinatorial and matrix techniques.

Findings

Constructed infinitely many new Cretan(4t+1) matrices

Established an inequality for the radius of these matrices

Provided a construction method for every order n ≥ 3

Abstract

A $Cretan(4t+1)$ matrix, of order $4t+1$, is an orthogonal matrix whose elements have moduli $\leq 1$. The only $Cretan(4t+1)$ matrices previously published are for orders 5, 9, 13, 17 and 37. This paper gives infinitely many new $Cretan(4t+1)$ matrices constructed using $regular~Hadamard$ matrices, $SBIBD(4t+1,k,\lambda)$, weighing matrices, generalized Hadamard matrices and the Kronecker product. We introduce an inequality for the radius and give a construction for a Cretan matrix for every order $n \geq 3$.