Subspace Codes in PG(2N - 1, Q)

An (r,M, 2δ; k)q constant–dimension subspace code, δ > 1, is a collection C of (k−1)–dimensional projective subspaces of PG(r−1, q) such that every (k−δ)–dimensional projective subspace of PG(r−1, q) is contained in at most a member of C. Constant–dimension subspace codes gained recently lot of interest due to the work by Koetter and Kschischang [18], where they presented an application of such codes for error-correction in random network coding. Here a (2n,M, 4;n)q constant–dimension subspace code is constructed, for every n ≥ 4. The size of our codes is considerably larger than all known constructions so far, whenever n > 4. When n = 4 a further improvement is provided by constructing an (8,M, 4; 4)q constant–dimension subspace code, with M = q + q(q + 1)(q + q + 1) + 1.