摘要 : Consider a simple, connected graph $\Gamma $ with n vertices. Let C be a code of length n with its coordinates corresponding to the vertices of $\Gamma $ . We define C as a storage code on $\Gamma $ if, for any codeword $c \... 展开 Consider a simple, connected graph $\Gamma $ with n vertices. Let C be a code of length n with its coordinates corresponding to the vertices of $\Gamma $ . We define C as a storage code on $\Gamma $ if, for any codeword $c \in C$ , the information at each coordinate of c can be recovered by accessing its neighboring coordinates. The main problem here is to construct high-rate storage codes on triangle-free graphs. In this paper, we employ the polynomial method to address a question proposed by Barg and Zémor in 2022, demonstrating that the BCH family of storage codes on triangle-free Cayley graphs achieves a unit rate. Furthermore, we generalize the construction of the BCH family and obtain more storage codes of unit rate on triangle-free graphs. We also compare the BCH family with the other known constructions by examining the rate of convergence of $1/(1-R(C_{n}))$ with respect to the length n, where $R(C_{n})$ is the rate of code $C_{n}$ . At last, we reveal a connection between the storage codes on triangle-free graphs and the Ramsey number $R(3,t)$ , which leads to an upper bound for the rate of convergence of $1/(1-R(C_{n}))$ . 收起