The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes

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As shown in [28], 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 [32,26] 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 expurgation-augmentation 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.