On Arcs and Quadrics

Abstract

An arc is a set of points of the $(k-1)$-dimensional projective space over the finite field with q elements ${\mathbb F}_q$, in which every k-subset spans the space. In this article, we firstly review Glynn’s construction of large arcs which are contained in the intersection of quadrics. Then, for q odd, we construct a series of matrices $\mathrm {F}_n$, where n is a non-negative integer and $n \leqslant |G|-k-1$, which depend on a small arc G. We prove that if G can be extended to a large arc S of size $q+2k-|G|+n-2$ then, for each vector v of weight three in the column space of $\mathrm {F}_n$, there is a quadric $\psi _v$ containing $S \setminus G$. This theorem is then used to deduce conditions for the existence of quadrics containing all the vectors of S.

The author acknowledges the support of the project MTM2014-54745-P of the Spanish Ministerio de Economía y Competitividad.

This article is for the proceedings of the International Workshop on the Arithmetic of Finite Fields WAIFI 2016, held in Ghent, Belgium, July 13–15, 2016.