Storage Codes on Coset Graphs with Asymptotically Unit Rate

A storage code on a graph G is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate converging to 3/4. Here we show that codes on such graphs can attain rate asymptotically approaching 1. Equivalently, this question can be phrased as a version of hat-guessing games on graphs (e.g., Cameron et al., in: Electron J Combin 23(1):48, 2016). In this language, we construct triangle-free graphs with success probability of the players approaching one as the number of vertices tends to infinity. Furthermore, finding linear index codes of rate approaching zero is also an equivalent problem.