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.

翻译：图$G$上的存储码是对顶点的一组符号分配，使得每个顶点可以通过观察其邻居恢复自身值。我们考虑在由二元线性码的陪集图构造的无三角形图上构建大尺寸存储码的问题。先前研究表明，存在无限族陪集图上的二元存储码，其速率收敛于3/4。本文证明此类图上的码可达到渐近趋近于1的速率。等价地，该问题可表述为图上帽子猜测游戏的一种版本（例如，P.J. Cameron等，《电子组合学杂志》2016年）。用此术语表述，我们构造了无三角形图，其中玩家成功概率随顶点数趋于无穷而趋近于1。此外，寻找速率趋近于零的线性索引码也是一个等价问题。另一族速率趋近于1的无三角形图存储码由A. Golovnev和I. Haviv（第36届计算复杂性会议，2021年）基于不同图族构建。