An Asymptotically Optimal Bound for Covering Arrays of Higher Index

A \emph{covering array} is an $N \times k$ array ($N$ rows, $k$ columns) with each entry from a $v$-ary alphabet, and for every $N\times t$ subarray, all $v^t$ tuples of size $t$ appear at least $\lambda$ times. The \emph{covering array number} is the smallest number $N$ for which such an array exists. For $\lambda = 1$, the covering array number is asymptotically logarithmic in $k$, when $v, t$ are fixed. Godbole, Skipper, and Sunley proved a bound of the form $\log k + \lambda \log \log k$ for the covering array number for arbitrary $\lambda$ and $v,t$ constant. The author proved a similar bound via a different technique, and conjectured that the $\log \log k$ term can be removed. In this short note we answer the conjecture in the affirmative with an asymptotically tight upper bound. In particular, we employ the probabilistic method in conjunction with the Lambert $W$ function.