The asymptotic behavior of Grassmannian codes

TL;DR

This paper analyzes the asymptotic behavior of Grassmannian codes, showing that known bounds are tight in the limit and determining the growth rates of optimal codes as dimensions grow.

Contribution

It proves that the iterated Johnson and Schönheim bounds are asymptotically attained for fixed parameters, and determines the asymptotics of optimal code sizes as dimensions increase.

Findings

Bounds are asymptotically tight for fixed parameters.

Asymptotic sizes of optimal codes are characterized.

Results apply as the ambient dimension tends to infinity.

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 Sch\"{o}nheim 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.