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.

图 G 上的存储代码是一组对顶点的符号分配，以便每个顶点都可以通过查看其邻居来恢复其值。我们考虑在构造为二进制线性码陪集图的无三角形图上构造大尺寸存储码的问题。之前已经表明，陪集图上存在无限族的二进制存储代码，其速率收敛于 3/4。在这里，我们表明，此类图上的代码可以达到渐近接近 1 的速率。同样，这个问题可以表述为图上猜帽子游戏的一个版本（例如，Cameron 等人，在：Electron J Combin 23(1) 中： 48，2016）。在这种语言中，我们构建了无三角形图，随着顶点数量趋于无穷大，玩家的成功概率接近 1。此外，寻找速率接近于零的线性索引码也是一个等价的问题。