Hadamard matrices related to the projective planes

TL;DR

This paper establishes a deep connection between the existence of projective planes of a given order and the construction of certain Hadamard matrices, revealing new equivalences in combinatorial design theory.

Contribution

It proves that a projective plane of order n exists if and only if a balancedly multi-splittable quaternary Hadamard matrix of order n^2 exists, linking two major combinatorial structures.

Findings

Existence of projective plane of order n is equivalent to a special Hadamard matrix of order n^2.

Introduces the concept of balancedly multi-splittable Hadamard matrices.

Provides a new characterization of projective planes through Hadamard matrices.

Abstract

Let $n$ be the order of a (quaternary) Hadamard matrix. It is shown that the existence of a projective plane of order $n$ is equivalent to the existence of a balancedly multi-splittable (quaternary) Hadamard matrix of order $n^2$.