On the finite geometry of $W(23,16)$

TL;DR

This paper investigates the geometric structure of the zero pattern of a hypothetical weighing matrix W(23,16), revealing properties that suggest such a matrix may not exist.

Contribution

The study characterizes the local geometry of W(23,16)'s zero pattern and constructs a related weighted graph, providing evidence against the existence of this weighing matrix.

Findings

More than 50% of line pairs intersect at a single point

Constructed a regular weighted graph from the geometry

Indications that W(23,16) may not exist

Abstract

We study the local geometry of the zero pattern of a weighing matrix $W(23,16)$. The geometry consists of $23$ lines and $23$ points where each line contains $7$ points. The incidence rules are that every two lines intersect in an odd number of points, and the dual statement holds as well. We show that more than $50\%$ of the pairs of lines must intersect at a single point, and construct a regular weighted graph out of this geometry. This might indicate that a weighing matrix $W(23,16)$ does not exist.