A graphical representation of binary linear codes

TL;DR

This paper introduces a graphical framework for binary linear codes by exploring the partial order of their cosets and constructing a graph based on the code's structure, providing new insights into code representations.

Contribution

It presents a novel graph construction for binary linear codes using the partial order of cosets, offering a new perspective on code structure analysis.

Findings

Established a partial order on cosets of binary linear codes

Constructed a graph based on the orphan structure of the code

Provides a new graphical representation for code analysis

Abstract

A binary $[n,k]$-linear code $\mathcal{C}$ is a $k$-dimensional subspace of $\mathbb{F}_2^n$. For $\boldsymbol{x}\in \mathbb{F}_2^n$, the set $\boldsymbol{x}+\mathcal{C}$ 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.