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.

翻译：设$\Gamma$为$n$个顶点上的简单连通图，$C$为长度为$n$的码字，其坐标由$\Gamma$的顶点索引。若对于任意码字$c \in C$，可通过访问$c$在$\Gamma$中的邻居恢复各坐标信息，则称$C$为$\Gamma$上的\textit{存储码}。本文的核心问题是在三角形无图上构造高码率存储码。我们解决了Barg与Z\'emor于2022年提出的一个开放问题，证明了BCH类存储码具有单位速率。此外，我们还推广了BCH类存储码的构造方法，在三角形无图上获得了更多具有单位速率的存储码。