Conference matrices with maximum excess and two-intersection sets

TL;DR

This paper constructs conference matrices with maximum excess using novel two-intersection sets derived from finite field block designs, advancing combinatorial design theory.

Contribution

It introduces a new class of two-intersection sets in finite field designs and applies them to construct conference matrices with maximum excess.

Findings

Existence of specific two-intersection sets in finite field designs.

Construction of conference matrices with maximum excess.

Link between two-intersection sets and conference matrix properties.

Abstract

A two-intersection set with parameters $(j;\alpha,\beta)$ for a block design is a $j$-subset of the point set of the design, which intersects every block in $\alpha$ or $\beta$ points. In this paper, we show the existence of a two-intersection set with parameters $(2m^2-m+1;m^2-m,m^2)$ for the block design obtained from translations of the set of nonzero squares in the finite field of order $q=4m^2+1$. As an application, we give a construction of conference matrices with maximum excess based on the two-intersection sets.