q-Analogs of Packing Designs

Abstract

A $P_q(t,k,n)$ $q$-packing design is a selection of $k$-subspaces of $\F_q^n$ such that each $t$-subspace is contained in at most one element of the collection. A successful approach adopted from the Kramer-Mesner-method of prescribing a group of automorphisms was applied by Kohnert and Kurz to construct some constant dimension codes with moderate parameters which arise by $q$-packing designs. In this paper we recall this approach and give a version of the Kramer-Mesner-method breaking the condition that the whole $q$-packing design must admit the prescribed group of automorphisms. Afterwards, we describe the basic idea of an algorithm to tackle the integer linear optimization problems representing the $q$-packing design construction by means of a metaheuristic approach. Finally, we give some improvements on the size of $P_2(2,3,n)$ $q$-packing designs.