Q‐Analogs of Packing Designs. Issue 7 (15th October 2013) - Braun, Michael; Reichelt, Jan
Abstract A <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg5cg04k9c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Pq(t, k, n)</alternatives>q‐packing design is a selection of k‐dimensional subspaces of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg5cg04k8t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Fqn</alternatives> 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 of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg5cg04kcg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />P2(2, 3, n)</alternatives>q‐packing designs.