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arXiv · 1603.05795

Extending small arcs to large arcs

Simeon Ball
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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.

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Simeon Ball. 2016-03-18. Extending small arcs to large arcs. https://arxiv.org/abs/1603.05795

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