The asymptotic behavior of Grassmannian codes

Abstract

The iterated Johnson bound is the best known upper bound on a size of an error-correcting code in the Grassmannian ${\mathcal{G}}_q(n,k)$. The iterated Schönheim bound is the best known lower bound on the size of a covering code in ${\mathcal{G}}_q(n,k)$. We use probabilistic methods to prove that both bounds are asymptotically attained for fixed $k$ and fixed radius, as $n$ approaches infinity. 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.

5 pages