  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.
