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 𝔽q, 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 Fn,wheren is a non-negative integer and n ≤ |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 Fn,there is a quadric ψv containing S\G. This theorem is then used to deduce conditions for the existence of quadrics containing all the vectors of S.