In [1], we introduced a family of combinatorial designs, which we call "alphabet reduction pairs of arrays", ARPAs for short. These designs depend on three integer parameters $q, p \leq q, k\leq p$: $q$ is the size of the symbol set $\{0, 1 ,\ldots, q -1\}$ in which the coefficients of the arrays take their values; $p$ is the maximum number of distinct symbols allowed in a row of the second array of the pair; $k$ is the larger integer for which the two arrays of the pair coincide -- up to the order of their rows -- on any $k$-ary subset of their columns. The first array must contain at least one occurrence of the word $0\ 1\ \ldots\ q -1$. Intuitively, the idea is to cover "as many as possible" occurrences of this word of $q$ symbols with "as few as possible" words of at most $p$ different symbols. These designs are related to the approximability of "Constraint Satisfaction Problems with bounded constraint arity", known as $k\,$CSPs. In this context, we are particularly interested in ARPAs in which the frequency of the word $0\ 1\ \ldots\ q -1$ is maximal. We introduce a seemingly simpler family of combinatorial designs as "Cover pairs of arrays" (CPAs). The arrays of a CPA take Boolean coefficients, and must still coincide on any $k$-ary subset of their columns. The purpose is, as it were, to cover "as many as possible" occurrences of the word of $q$ ones using "as few as possible" $q$-length Boolean words of weight at most $p$. We show that, when it comes to maximizing the frequency of the words $0\ 1\ \ldots\ q -1$ in ARPAs and $1\ 1\ \ldots\ 1$ in CPAs, ARPAs and CPAs are equivalent. We prove the optimality of the ARPAs given in [1] for the case $p =k$. In addition, we provide optimal ARPAs for the cases $k =1$ and $k =2$. We emphasize the fact that both families of combinatorial designs are related to the approximability of $k\,$CSPs.

翻译：在文献[1]中，我们引入了一族组合设计，称为“阵列字母表缩减对”（简称ARPAs）。这类设计依赖于三个整数参数 $q, p \leq q, k\leq p$：$q$ 表示符号集 $\{0, 1 ,\ldots, q -1\}$ 的大小，阵列的系数在该集合中取值；$p$ 表示配对中第二个阵列的任一行所允许的最大不同符号数；$k$ 是使得配对中的两个阵列在任意 $k$ 元列子集上（至多行的顺序不同）保持一致的最大整数。第一个阵列必须至少包含一次词 $0\ 1\ \ldots\ q -1$。直观上，其核心思想是用“尽可能少”的、至多包含 $p$ 种不同符号的词，去覆盖“尽可能多”的出现次数的这个 $q$ 符号词。这类设计与“约束元数有界的约束满足问题”（即 $k\,$CSPs）的近似性相关。在此背景下，我们特别关注词 $0\ 1\ \ldots\ q -1$ 出现频率最大的 ARPAs。我们引入了另一族看似更简单的组合设计，称为“覆盖阵列对”（CPAs）。CPA 中的阵列采用布尔系数，并且仍然需要在任意 $k$ 元列子集上保持一致。其目的可以说是用“尽可能少”的、权重至多为 $p$ 的 $q$ 长布尔词，去覆盖“尽可能多”的出现次数的全 $1$ 词（即 $1\ 1\ \ldots\ 1$）。我们证明，在最大化 ARPAs 中词 $0\ 1\ \ldots\ q -1$ 的频率与 CPAs 中词 $1\ 1\ \ldots\ 1$ 的频率这一问题上，ARPAs 与 CPAs 是等价的。我们证明了文献[1]中给出的 $p =k$ 情形的 ARPAs 的最优性。此外，我们为 $k =1$ 和 $k =2$ 的情形提供了最优的 ARPAs。我们强调，这两族组合设计均与 $k\,$CSPs 的近似性相关。