The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes

As shown in [31], one of the five isomorphism types of optimal binary subspace codes of size 77 for packet length v = 6, constant dimension k = 3 and minimum subspace distance d = 4 can be constructed by first expurgating and then augmenting the corresponding lifted Gabidulin code in a fairly simple way. The method was refined in [36, 29] to yield an essentially computer-free construction of a currently best-known plane subspace code of size 329 for (v, k, d) = (7, 3, 4). In this paper we generalize the expurgationaugmentation approach to arbitrary packet length v, providing both a detailed theoretical analysis of our method and computational results for small parameters. As it turns out, our method is capable of producing codes larger than those obtained by the echelon-Ferrers construction and its variants. We are able to prove this observation rigorously for packet lengths v ≡ 3 (mod 4).