On Sets with Few Intersection Numbers in Finite Projective and Affine Spaces

Nicola Durante

doi:10.37236/3434

Abstract

In this paper we study sets $X$ of points of both affine and projective spaces over the Galois field $\mathop{\rm{GF}}(q)$ such that every line of the geometry that is neither contained in $X$ nor disjoint from $X$ meets the set $X$ in a constant number of points and we determine all such sets. This study has its main motivation in connection with a recent study of neighbour transitive codes in Johnson graphs by Liebler and Praeger [Designs, Codes and Crypt., 2014]. We prove that, up to complements, in $\mathop{\rm{PG}}(n,q)$ such a set $X$ is either a subspace or $n=2,q$ is even and $X$ is a maximal arc of degree $m$. In $\mathop{\rm{AG}}(n,q)$ we show that $X$ is either the union of parallel hyperplanes or a cylinder with base a maximal arc of degree $m$ (or the complement of a maximal arc) or a cylinder with base the projection of a quadric. Finally we show that in the affine case there are examples (different from subspaces or their complements) in $\mathop{\rm{AG}}(n,4)$ and in $\mathop{\rm{AG}}(n,16)$ giving new neighbour transitive codes in Johnson graphs.

Highlights

Let AG(n, q), q power of a prime number, be the affine space of dimension n (n 2) over the Galois field GF(q), let Σ∞ be the (n − 1)-dimensional projective space of its points at infinity and let PG(n, q) = AG(n, q) ∪ Σ∞ its projective completion, the projective space of dimension n over GF(q).Let m1 < m2 < . . . < mt be non-negative integers
In this paper we study sets X of points of both affine and projective spaces over the Galois field GF(q) such that every line of the geometry that is neither contained in X nor disjoint from X meets the set X in a constant number of points and we determine all such sets
This study has its main motivation in connection with a recent study of neighbour transitive codes in Johnson graphs by Liebler and Praeger [Designs, Codes and Crypt., 2014]

Summary

Preliminaries

Among the known sets with few intersection numbers some with external lines or containing lines in PG(n, q) (or AG(n, q)) are: subspaces, quadrics, Hermitian varieties, subgeometries, maximal arcs in PG(2, q) (or AG(2, q)), q even, internal points of conics in PG(2, q) (or AG(n, q)), q odd or their complements Note that none of the sets mentioned up to now, except subspaces and their complements, contain both a line and have an external line and all but m-maximal arcs and internal points of a conic have at least a 1-secant line. We will give examples of sets of points with one proper intersection number, different from 1, containing lines and having external lines in affine spaces. Q in exactly one point (see [14], [18])

Motivations

Sets with two intersection numbers in projective and affine spaces

The projective case

The affine case