Extending small arcs to large arcs

TL;DR

This paper establishes bounds on the size of large arcs in finite vector spaces of odd characteristic, using matrix properties, and explores their classification and construction.

Contribution

It introduces a novel approach to bounding and classifying large arcs via matrix analysis, advancing understanding in finite geometry.

Findings

Derived upper bounds on large arcs in odd characteristic spaces.

Identified conditions for the largest arcs containing a given arc.

Provided tools for computational classification and construction of large arcs.

Abstract

An arc is a set of vectors of the $k$-dimensional vector space over the finite field with $q$ elements ${\mathbb F}_q$, in which every subset of size $k$ is a basis of the space, i.e. every $k$-subset is a set of linearly independent vectors. Given an arc $G$ in a space of odd characteristic, we prove that there is an upper bound on the largest arc containing $G$. The bound is not an explicit bound but is obtained by computing properties of a matrix constructed from $G$. In some cases we can also determine the largest arc containing $G$, or at least determine the hyperplanes which contain exactly $k-2$ vectors of the large arc. The theorems contained in this article may provide new tools in the computational classification and construction of large arcs.