On the Bounds of Certain Maximal Linear Codes in a Projective Space
B. Srikanth Pai, Balaji Sundar Rajan · IEEE Transactions on Information Theory · 2015
The set of all subspaces of Fqnis denoted by Pq(n). The subspace distance dS(X, Y) = dim(X) + dim(Y) - 2 dim(X ∩ Y) defined on Pq(n) turns it into a natural coding space for error correction in random network coding. A subset of Pq(n) is called a code and the subspaces that belong to the code are called codewords. Motivated by classical coding theory, a linear coding structure can be imposed on a subset of Pq(n). Braun et al. conjectured that the largest cardinality of a linear code, that contains Fqn, is 2n. In this paper, we prove this conjecture and characterize the maximal linear codes that contain Fqn.