  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.,
P.J. Cameron e.a., \emph{Electronic J. Comb.} 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. Another family of storage codes on triangle-free graphs of rate
approaching 1 was constructed earlier by A. Golovnev and I. Haviv (36th
Computational Complexity Conf., 2021) relying on a different family of graphs.
