Two-level Cretan Matrices Constructed Theoretically and Computationally using SBIBD

TL;DR

This paper explores the construction of Cretan matrices using both theoretical and computational methods, highlighting their properties and potential applications, especially through incidence matrices and difference sets.

Contribution

It introduces new constructions of Cretan matrices via SBIBD and Hadamard difference sets, comparing theoretical and computational approaches.

Findings

Cretan matrices can be constructed from SBIBD incidence matrices.

Hadamard difference sets enable new Cretan matrix families.

Theoretical and computational methods yield complementary results.

Abstract

Cretan matrices are orthogonal matrices with elements $\leq 1$. These may have application in forming some new materials. There is a search for Cretan matrices, especially with high determinant, for all orders. These have been found by both mathematical and computational methods. This paper highlights the differences between theoretical and computational solutions to finding Cretan matrices. It has been shown that the incidence matrix of a symmetric balanced incomplete block design can be used to form Cretan($v;2$) matrices. We give families of Cretan matrices constructed using Hadamard related difference sets.