  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 $\mathbb{F}_q^n$.
