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

TL;DR

This paper proves Braun, Etzion, and Vardy's conjecture that the largest linear code in a projective space containing the entire space has size 2^n, and characterizes these maximal codes.

Contribution

It confirms the conjecture about the maximum size of linear codes in projective spaces and characterizes the structure of maximal such codes.

Findings

Largest linear code size is 2^n.

Maximal linear codes containing the full space are characterized.

Conjecture by Braun, Etzion, and Vardy is proven.

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 $\mathbb{F}_q^n$.