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.