q ‐Analogs of Packing Designs

AP q ( t , k , n )q ‐packing design is a selection of k ‐dimensional subspaces of F q n such that each t ‐dimensional 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 that 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 ofP 2 ( 2 , 3 , n )q ‐packing designs.