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 Schönheim 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.