On the Bounds of Certain Maximal Linear Codes in a Projective Space

The set of all subspaces of F_qⁿ is denoted by P_q(n). The subspace distance d_S(X, Y) = dim(X) + dim(Y) - 2 dim(X ∩ Y) defined on P_q(n) turns it into a natural coding space for error correction in random network coding. A subset of P_q(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 P_q(n). Braun et al. conjectured that the largest cardinality of a linear code, that contains F_qⁿ, is 2ⁿ. In this paper, we prove this conjecture and characterize the maximal linear codes that contain F_qⁿ.