Non-transversal Vectors of Some Finite Geometries

TL;DR

This paper introduces the concept of non-transversal vectors in finite geometries, leading to efficient constructions of certain biplanes and revealing structural dichotomies based on incidence matrix symmetry.

Contribution

It defines non-transversal vectors and uses them to construct four biplanes of order 9, improving understanding of their structural properties.

Findings

Constructed four biplanes of order 9 using new invariants

Identified a structural dichotomy in biplanes of orders 7 and 9

Provided more efficient methods for finite geometry constructions

Abstract

By means of associated structural invariants, we efficiently construct four biplanes of order 9 - except the one with the smallest automorphism group, that is found by Janko and Trung. The notion of non-transversal vector is introduced since we observed related properties that provide significantly more efficient constructions. There is a dichotomy in the structure of biplanes of order 7 and 9 with respect to the incidence matrix symmetry.