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

The set of all subspaces of $\mathbb{F}_q^n$ is denoted by $\mathbb{P}_q(n)$. The subspace distance $d_S(X,Y) = \dim(X)+ \dim(Y) - 2\dim(X \cap Y)$ defined on $\mathbb{P}_q(n)$ turns it into a natural coding space for error correction in random network coding. A subset of $\mathbb{P}_q(n)$ 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 $\mathbb{P}_q(n)$. Braun, Etzion and Vardy conjectured that the largest cardinality of a linear code, that contains $\mathbb{F}_q^n$, is $2^n$. In this paper, we prove this conjecture and characterize the maximal linear codes that contain $\mathbb{F}_q^n$.