A graphical representation of binary linear codes

Abstract

A binary $[n,k]$-linear code $\mathcal{C}$ is a $k$-dimensional subspace of ${\mathbb{F}}_2^n$. For $\mathbf{x}\in {\mathbb{F}}_2^n$, the set $\mathbf{x}+\mathcal{C}$ is a coset of $\mathcal{C}$. In this work we study a partial ordering on the set of cosets of a binary linear code $\mathcal{C}$ of length $n$ and we construct a graph using the orphan structure of this code.