Storage codes on graphs are an instance of \emph{codes with locality}, which are used in distributed storage schemes to provide local repairability. Specifically, the nodes of the graph correspond to storage servers, and the neighbourhood of each server constitute the set of servers it can query to repair its stored data in the event of a failure. A storage code on a graph with $n$-vertices is a set of $n$-length codewords over $\field_q$ where the $i$th codeword symbol is stored in server $i$, and it can be recovered by querying the neighbours of server $i$ according to the underlying graph. In this work, we look at binary storage codes whose repair function is the parity check, and characterise the tradeoff between the locality of the code and its rate. Specifically, we show that the maximum rate of a code on $n$ vertices with locality $r$ is bounded between $1-1/n\lceil n/(r+1)\rceil$ and $1-1/n\lceil n/(r+1)\rceil$. The lower bound on the rate is derived by constructing an explicit family of graphs with locality $r$, while the upper bound is obtained via a lower bound on the binary-field rank of a class of symmetric binary matrices. Our upper bound on maximal rate of a storage code matches the upper bound on the larger class of codes with locality derived by Tamo and Barg. As a corollary to our result, we obtain the following asymptotic separation result: given a sequence $r(n), n\geq 1$, there exists a sequence of graphs on $n$-vertices with storage codes of rate $1-o(1)$ if and only if $r(n)=\omega(1)$.

翻译：图中的存储码是具有局部性的编码实例，用于分布式存储方案中提供局部可修复性。具体而言，图的顶点对应存储服务器，每个服务器的邻居构成其在故障时可通过查询修复所存数据的服务器集合。在n个顶点的图上，存储码是定义在域𝔽_q上的一组n长码字，其中第i个码字符号存储于服务器i，且可通过根据底层图查询服务器i的邻居来恢复该符号。本文研究修复函数为奇偶校验的二元存储码，并刻画编码局部性与码率之间的权衡关系。具体而言，我们证明n个顶点上局部性为r的编码的最大码率介于1-1/n⌈n/(r+1)⌉与1-1/n⌈n/(r+1)⌉之间。码率下界通过构造显式的局部性为r的图族得到，而上界则通过一类对称二元矩阵的二元域秩的下界导出。本文关于存储码最大码率的上界与Tamo和Barg推导的更大类局部性编码的上界一致。作为结果的推论，我们得到如下渐近分离结果：给定序列r(n), n≥1，存在n顶点图序列，其存储码码率为1-o(1)当且仅当r(n)=ω(1)。