On sets with few intersection numbers in finite projective and affine spaces
Summary: We study sets \(X\) of points of both affine and projective spaces over the Galois field \(\mathrm{GF}(q)\) 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 \textit{R. A. Liebler} and \textit{C. E. Praeger} [Des. Codes Cryptography 73, No. 1, 1--25 (2014; Zbl 1296.05089)]. We prove that, up to complements, in \(\mathrm{PG}(n,q)\) such a set \(X\) is either a subspace or \(n=2,q\) is even and \(X\) is a maximal arc of degree \(m\). In \(\mathrm{AG}(n,q)\) 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 \(\mathrm{AG}(n,4)\) and in \(\mathrm{AG}(n,16)\) giving new neighbour transitive codes in Johnson graphs.
scientific article; zbMATH DE number 6360688
scientific article