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

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.