A \emph{covering array} is an $N \times k$ array of elements from a $v$-ary alphabet such that every $N \times t$ subarray contains all $v^t$ tuples from the alphabet of size $t$ at least $\lambda$ times; this is denoted as $\CA_\lambda(N; t, k, v)$. Covering arrays have applications in the testing of large-scale complex systems; in systems that are nondeterministic, increasing $\lambda$ gives greater confidence in the system's correctness. The \emph{covering array number}, $\CAN_\lambda(t,k,v)$ is the smallest number of rows for which a covering array on the other parameters exists. For general $\lambda$, only several nontrivial bounds are known, the smallest of which was asymptotically $\log k + \lambda \log \log k + o(\lambda)$ when $v, t$ are fixed. Additionally it has been conjectured that the $\log \log k$ term can be removed. First, we affirm the conjecture by deriving an asymptotically optimal bound for $\CAN_\lambda(t,k,v)$ for general $\lambda$ and when $v, t$ are constant using the Stein--Lov\'asz--Johnson paradigm. Second, we improve upon the constants of this method using the Lov\'asz local lemma. Third, when $\lambda=2$, we extend a two-stage paradigm of Sarkar and Colbourn that improves on the general bound and often produces better bounds than even when $\lambda=1$ of other results. Fourth, we extend this two-stage paradigm further for general $\lambda$ to obtain an even stronger upper bound, including using graph coloring. And finally, we determine a bound on how large $\lambda$ can be for when the number of rows is fixed.

翻译：覆盖数组是一个$N \times k$的数组，其元素取自一个$v$元字母表，要求每个$N \times t$子数组至少包含$\lambda$次所有$v^t$个长度为$t$的字母表元组；记为$\CA_\lambda(N; t, k, v)$。覆盖数组在大型复杂系统的测试中有重要应用；对于非确定性系统，增大$\lambda$可提高对系统正确性的置信度。覆盖数组数$\CAN_\lambda(t,k,v)$是指存在其他参数覆盖数组的最少行数。对于一般$\lambda$，目前仅已知几个非平凡界，其中当$v, t$固定时，最小者渐近为$\log k + \lambda \log \log k + o(\lambda)$。此外，有猜想认为$\log \log k$项可以去掉。首先，我们利用Stein–Lovász–Johnson范式证实了该猜想，导出了当$v, t$为常数时一般$\lambda$下$\CAN_\lambda(t,k,v)$的渐近最优界。其次，我们利用Lovász局部引理改进了该方法的常数。第三，当$\lambda=2$时，我们扩展了Sarkar和Colbourn的两阶段范式，该范式改进了一般界，且通常能产生比$\lambda=1$时其他结果更优的界。第四，我们进一步将这一两阶段范式推广至一般$\lambda$，包括利用图着色方法，得到了更强的上界。最后，我们确定了当行数固定时$\lambda$所能达到的最大值的界。