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

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.

Characterizations of Finite Projective and Affine Spaces

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.Theorem 1.A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4if and only if there are positive integers v, k, and y, with μ> 1and(μ– l)(v — k) ≠ (k—μ)2such that the following assumptions hold.(I)Every block is on k points, and every two intersecting blocks are on μ common points.(II)Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks.(III)Given two distinct points p and q, there is a block on p but not on q.(IV)There are v points, and v– 2 ≧k>μ.