报告地点：行健楼学术活动室526
邀请人：周海燕教授
摘要：Consider a simple, connected graph Γ with 𝑛 vertices. Let 𝐶 be a code of length 𝑛 with its coordinates corresponding to the vertices of Γ. We de ne 𝐶 as a storage code on Γ if, for any codeword 𝑐 ∈ 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 asked by Barg and Z ́emor 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(𝐶𝑛)) with respect to the length 𝑛, where 𝑅(𝐶𝑛) is the rate of code 𝐶𝑛. 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(𝐶𝑛)).
