The BCH Family of Storage Codes on Triangle-Free Graphs is of Unit Rate

TL;DR

This paper proves that the BCH family of storage codes on triangle-free graphs achieves unit rate and introduces generalized constructions for such high-rate codes.

Contribution

It solves an open problem by showing BCH codes have unit rate on triangle-free graphs and generalizes the construction for more such codes.

Findings

BCH family of storage codes has unit rate on triangle-free graphs

Generalized construction yields more high-rate storage codes

Addresses an open problem from 2022

Abstract

Let $\Gamma$ be a simple connected graph on $n$ vertices, and let $C$ be a code of length $n$ whose coordinates are indexed by the vertices of $\Gamma$. We say that $C$ is a \textit{storage code} on $\Gamma$ if for any codeword $c \in C$, one can recover the information on each coordinate of $c$ by accessing its neighbors in $\Gamma$. The main problem here is to construct high-rate storage codes on triangle-free graphs. In this paper, we solve an open problem posed by Barg and Z\'emor in 2022, showing that the BCH family of storage codes is of unit rate. Furthermore, we generalize the construction of the BCH family and obtain more storage codes of unit rate on triangle-free graphs.