The Asymptotic Behavior of Grassmannian Codes
The iterated Johnson bound is the best known upper bound on the size of an error-correcting code in the Grassmannian G(q)(n, k). The iterated Schonheim bound is the best known lower bound on the size of a covering code in G(q)(n, k). We prove that both bounds are asymptotically attained for fixed k and fixed radius, as n approaches infinity. Our methods rely on results from the theory of quasi-random hypergraphs which are proved using probabilistic techniques. We also determine the asymptotics of the size of the best Grassmannian codes and covering codes when n - k and the radius are fixed, as n approaches infinity.
grassmannian hypergraph constant dimension code covering bound packing bound