  We consider the problem of merging two sorted sequences on a comparator
network that is used repeatedly, that is, if the output is not sorted, the
network is applied again using the output as input. The challenging task is to
construct such networks of small depth (called a period in this context). In
our previous paper entitled Faster 3-Periodic Merging Network we reduced twice
the time of merging on $3$-periodic networks, i.e. from $12\log N$ to $6\log
N$, compared to the first construction given by Kuty{\l}owski, Lory\'s and
Oesterdikhoff. Note that merging on $2$-periodic networks require linear time.
In this paper we extend our construction, which is based on Canfield and
Williamson $(\log N)$-periodic sorter, and the analysis from that paper to any
period $p \ge 4$. For $p\ge 4$ our $p$-periodic network merges two sorted
sequences of length $N/2$ in at most $\frac{2p}{p-2}\log N + p\frac{p-8}{p-2}$
rounds. The previous bound given by Kuty{\l}owski at al. was
$\frac{2.25p}{p-2.42}\log N$. That means, for example, that our $4$-periodic
merging networks work in time upper-bounded by $4\log N$ and our $6$-periodic
ones in time upper-bounded by $3\log N$ compared to the corresponding $5.67\log
N$ and $3.8\log N$ previous bounds. Our construction is regular and follows the
same periodification schema, whereas some additional techniques were used
previously to tune the construction for $p \ge 4$. Moreover, our networks are
also periodic sorters and tests on random permutations show that average
sorting time is closed to $\log^2 N$.
