  Boolean cardinality constraints state that at most (at least, or exactly) $k$
out of $n$ propositional literals can be true. We propose a new class of
selection networks that can be used for an efficient encoding of them. Several
comparator networks have been proposed recently for encoding cardinality
constraints and experiments have proved their efficiency. Those were based
mainly on the odd-even or pairwise comparator networks. We use similar ideas,
but we extend the model of comparator networks so that the basic components are
not only comparators (2-sorters) but more general $m$-sorters, for $m \geq 2$.
The inputs are organized into $m$ columns, in which elements are recursively
selected and, after that, columns are merged using an idea of multi-way
merging. We present two algorithms parametrized by $m \geq 2$. We call those
networks $m$-Wise Selection Network and $m$-Odd-Even Selection Network. We give
detailed construction of the mergers when $m=4$. The construction can be
directly applied to any values of $k$ and $n$. The proposed encoding of sorters
is standard, therefore the arc-consistency is preserved. We prove correctness
of the constructions and present the theoretical and experimental evaluation,
which show that the new encodings are competitive to the other state-of-art
encodings.
