Construction of storage codes of rates approaching one on triangle-free graphs

Consider an assignment of bits to the vertices of a connected graph $\Gamma(V, E)$ with the property that the value of each vertex is a function of the values of its neighbors. A collection of such assignments is called a storage code of length $|V|$ on $\Gamma$. In this paper we construct an infinite family of binary linear storage codes on triangle-free graphs with rates arbitrarily close to one.