  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.
