A covering array $\mathrm{CA}(N;t,k,v)$ of strength $t$ is an $N \times k$ array of symbols from an alphabet of size $v$ such that in every $N \times t$ subarray, every $t$-tuple occurs in at least one row. A covering array is \emph{optimal} if it has the smallest possible $N$ for given $t$, $k$, and $v$, and \emph{uniform} if every symbol occurs $\lfloor N/v \rfloor$ or $\lceil N/v \rceil$ times in every column. Prior to this paper the only known optimal covering arrays for $t=2$ were orthogonal arrays, covering arrays with $v=2$ constructed from Sperner's Theorem and the Erdős-Ko-Rado Theorem, and eleven other parameter sets with $v>2$ and $N > v^2$. In all these cases, there is a uniform covering array with the optimal size. It has been conjectured that there exists a uniform covering array of optimal size for all parameters. In this paper a new lower bound as well as structural constraints for small uniform strength-$2$ covering arrays are given. Moreover, covering arrays with small parameters are studied computationally. The size of an optimal strength-$2$ covering array with $v > 2$ and $N > v^2$ is now known for $21$ parameter sets. Our constructive results continue to support the conjecture.

中文翻译：

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小强度-2覆盖阵列的结构

强度为 $t$ 的覆盖数组 $\mathrm{CA}(N;t,k,v)$ 是一个 $N \times k$ 符号数组，来自大小为 $v$ 的字母表，使得在每个 $N \次 t$ 子数组，每个 $t$-tuple 至少出现在一行中。如果对于给定的 $t$、$k$ 和 $v$，覆盖数组具有最小的可能 $N$，则覆盖数组是 \emph{optimal}，如果每个符号都出现 $\lfloor N/v\，则 \emph{uniform}每列中 rfloor$ 或 $\lceil N/v \rceil$ 次。在本文之前，对于 $t=2$ 唯一已知的最佳覆盖数组是正交数组，覆盖数组具有 $v=2$ 由 Sperner 定理和 Erdős-Ko-Rado 定理构造而成，以及其他 11 个参数集 $v> 2$ 和 $N > v^2$。在所有这些情况下，都有一个具有最佳尺寸的均匀覆盖阵列。已经推测，对于所有参数存在最优大小的均匀覆盖阵列。在本文中，给出了小均匀强度-$2$ 覆盖阵列的新下界和结构约束。此外，计算研究了具有小参数的覆盖阵列。$v > 2$ 和 $N > v^2$ 的最佳强度 $2$ 覆盖数组的大小现在已知 $21$ 参数集。我们的建设性结果继续支持这一猜想。