  A $t$-covering array with entries from the alphabet ${\cal
Q}=\{0,1,\ldots,q-1\}$ is a $k\times n$ stack, so that for any choice of $t$
(typically non-consecutive) columns, each of the $q^{t}$ possible $t$-letter
words over ${\cal Q}$ appear at least once among the rows of the selected
columns. We will show how a combination of the Lov\'asz local lemma;
combinatorial analysis; Stirling's formula; and Calculus enables one to find
better asymptotic bounds for the minimum size of $t$-covering arrays, notably
for $t = 3, 4$. Here size is measured in the number of rows, as expressed in
terms of the number of columns.
