Extending small arcs to 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.