We determine the maximum size A 2 (8, 6; 4) of a binary subspace code of packet length v = 8, minimum subspace distance d = 6, and constant dimension k = 4 to be 257. There are two isomorphism types of optimal codes. Both of them are extended LMRD codes. In finite geometry terms, the maximum number of solids in PG(7, 2) mutually intersecting in at most a point is 257. The result was obtained by combining the classification of substructures with integer linear programming techniques. This result implies that the maximum size A 2 (8, 6) of a binary mixed-dimension subspace code of packet length 8 and minimum subspace distance 6 is 257 as well.
We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance d = 4, dimension k = 3 of the codewords for all field sizes q, and sufficiently large dimensions v of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.
Abstract. One of the main problems of the research area of network coding is to compute good lower and upper bounds of the achievable cardinality of so-called subspace codes in Pq(n), i.e., the set of subspaces of F n q , for a given minimal distance. Here we generalize a construction of Etzion and Silberstein to a wide range of parameters. This construction, named coset construction, improves or attains several of the previously best-known subspace code sizes and attains the MRD bound for an infinite family of parameters.
We generalize upper bounds for constant dimension codes containing a lifted maximum rank distance code first studied by Etzion and Silberstein. The proof allows to construct several improved codes.
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