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

The set of all subspaces of F
_q
ⁿ
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
_q
ⁿ
, is 2
ⁿ
. In this paper, we prove this conjecture and characterize the maximal linear codes that contain F
_q
ⁿ
.