Isometry classes of codes arising from sets of points in the projective plane

Abstract

Consider the natural action of PGL 3( q) on the projective plane PG 2( q) over a finite field GF( q). In this paper we split a set of representatives of conjugacy classes of PGL 3( q) into a disjoint union of subfamilies gathering the elements that have the same cycle type as a permutation of the points of PG 2( q). Also, we count the number of elements in each subfamily. This allows us to obtain formulas for the number of orbits of the action of PGL 3( q) on different sets of n-subsets and n-multisubsets of PG 2( q). As an application we obtain explicit formulas for the number of isometry classes of different families of codes of dimension three.