The Asymptotic Behavior of Grassmannian Codes
Simon R. Blackburn⋆, Tuvi Etzion · IEEE Transactions on Information Theory · 2012
The iterated Johnson bound is the best known upper bound on the size of an error-correcting code in the GrassmannianGq(n,k). The iterated Schönheim bound is the best known lower bound on the size of a covering code inGq(n,k). We prove that both bounds are asymptotically attained for fixedkand fixed radius, asnapproaches 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 whenn-kand the radius are fixed, asnapproaches infinity.