On linear subspace codes closed under intersection

Subspace codes are subsets of the projective space P_q(n), which is the set of all subspaces of the vector space F_qⁿ. Koetter and Kschischang argued that subspace codes are useful for error and erasure correction in random network coding. Linearity in subspace codes was defined by Braun, Etzion and Vardy, and they conjectured that the largest cardinality of a linear subspace code in P_q(n) is 2ⁿ. In this paper, we show that the conjecture holds for linear subspace codes that are closed under intersection, i.e., codes having the property that the intersection of any pair of codewords is also a codeword. The proof is via a characterization of such codes in terms of partitions of linearly independent subsets of F_qⁿ.