  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.
