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

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.

Neighbour-transitive codes in Johnson graphs

“…As a broad summary of the results of this paper, together with those of [10] and [21], for the case of minimum distance at least 3, we can announce that:…”

Codes with few weights arising from linear sets

“…Despite of the number of the algebraic constructions for three-weight codes, see e.g. [13,14,15,33,36], there are few known geometric constructions [19].…”

Sporadic neighbour-transitive codes in Johnson graphs

“…By [12, Proposition 7.4], either (i) the code consists of subspaces or their complements of some fixed dimension, or (ii) each codeword is a subset of points of PG(n − 1, q) of class [0, x, q + 1] 1 (defined as above), where x = 2, or q = q 2 0 and x = q 0 + 1. In this case, Durante [6, Theorem 3.2] drew together results about subsets of PG(n − 1, q) of class [0, x, q + 1] 1 , and showed in [6,Theorem 3.3] that no such subsets, apart from subspaces and their complements, have the symmetry property required for strongly incidence-transitive codes.…”