q-Analogs of Packing Designs

TL;DR

This paper explores methods for constructing $q$-packing designs, extending existing algorithms with metaheuristics, and presents improved sizes for specific $q$-packing design parameters, advancing combinatorial design theory.

Contribution

It introduces a flexible version of the Kramer-Mesner-method and applies metaheuristic algorithms to optimize $q$-packing design construction.

Findings

Improved sizes for $P_2(2,3,n)$ $q$-packing designs.

Extended the Kramer-Mesner-method to allow non-automorphism-preserving designs.

Applied metaheuristics to solve integer linear optimization problems in design construction.

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.