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

Abstract

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 ${\mathbb{F}}_q^n$.

10 pages, no figures