Pseudo-Hadamard matrices of the first generation and an algorithm for producing them

TL;DR

This paper introduces pseudo-Hadamard matrices of the first generation, extending the classical concept, and presents an algorithm for generating them to test related mathematical conjectures.

Contribution

It defines pseudo-Hadamard matrices of the first generation and develops an algorithm for their generation, enabling testing of new mathematical conjectures.

Findings

Defined pseudo-Hadamard matrices of the first generation

Developed an algorithm for generating these matrices

Used the algorithm to test mathematical conjectures

Abstract

Hadamard matrices in $\{0,1\}$ presentation are square $m\times m$ matrices whose entries are zeros and ones and whose rows considered as vectors in $\Bbb R^m$ produce the Gram matrix of a special form with respect to the standard scalar product in $\Bbb R^m$. The concept of Hadamard matrices is extended in the present paper. As a result pseudo-Hadamard matrices of the first generation are defined and investigated. An algorithm for generating these pseudo-Hadamard matrices is designed and is used for testing some conjectures.