A Note on the Asymptotic Least Density of Covering Codes in $[q]^n$

TL;DR

This paper improves the upper bound on the asymptotic least density of covering codes in $[q]^n$ by refining the optimization approach used in prior work.

Contribution

It introduces a modified optimization technique that yields a constant factor improvement over existing upper bounds for covering codes.

Findings

Achieved a tighter upper bound on covering code density.

Demonstrated the effectiveness of a different optimization method.

Provided insights into the asymptotic behavior of covering codes.

Abstract

In this short note we revisit the upper bound of the asymptotic least density of covering codes of radius $R$ in $[q]^n$ established by Krivelevich, Sudakov, and Vu. We show that by using a slightly different optimization in their core theorem we can obtain a constant factor improvement to their upper bound.