Extending small arcs to large arcs

An arc is a set of vectors of the k -dimensional vector space over the finite field with q elements 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.