On the bounds of certain maximal linear codes in a projective space

The set of all subspaces of F
ⁿ
q
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, Etzion and Vardy conjectured that the largest cardinality of a linear code, that contains F
ⁿ
q
, is 2
ⁿ
. In this paper, we prove this conjecture and characterize the maximal linear codes that contain F
ⁿ
q
.