Constructions and bounds for mixed-dimension subspace codes

Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. The resulting so-called Main Problem of Subspace Coding is to determine the maximum size Aq(v, d) of a code in PG(v−1,Fq) with minimum subspace distance d. Here we completely resolve this problem for d ≥ v − 1. For d = v − 2 we present some improved bounds and determine Aq(5, 3) = 2q3 + 2 (all q), A2(7, 5) = 34. We also provide an exposition of the known determination of Aq(v, 2), and a table with exact results and bounds for the numbers A2(v, d), v ≤ 7.