Abstract：
Let Γ be a simple graph with n vertices and C a code of length n with coordinates indexed by the vertices of Γ. Say C is a storage code on Γ if for each codeword c, every coordinate can be recovered by its neighbours.
Constructing codes of high rate on triangle-free graphs represents a challenge. Recently Barg and Zemor presented an infinite family of storage codes of rate 3/4 on triangle-free graphs, it is called the Hamming family. We will introduce how to generalize this family to reach asymptotically unit rate. This wrok is inspired by cyclic code, which is treated as an ideal in a quotient polynomial ring.
