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

The set of all subspaces of \mathbb Fq^{n} is denoted by \mathbb Pq(n) . The subspace distance dS(X,Y) = \dim (X)+ \dim (Y)\, - 2\dim (X \cap Y) defined on \mathbb Pq(n) turns it into a natural coding space for error correction in random network coding. A subset of \mathbb P\vphantom {RRRq}(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 \mathbb Pq(n) . Braun et al. conjectured that the largest cardinality of a linear code, that contains \mathbb Fq^{n} , is 2^n . In this paper, we prove this conjecture and characterize the maximal linear codes that contain \mathbb Fq^{n} . [ABSTRACT FROM AUTHOR]