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

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.