  In this paper, we study the problem of computing the majority function by
low-depth monotone circuits and a related problem of constructing low-depth
sorting networks. We consider both the classical setting with elementary
operations of arity $2$ and the generalized setting with operations of arity
$k$, where $k$ is a parameter. For both problems and both settings, there are
various constructions known, the minimal known depth being logarithmic.
However, there is currently no known construction that simultaneously achieves
sub-log-squared depth, effective constructability, simplicity, and has a
potential to be used in practice. In this paper we make progress towards
resolution of this problem.
  For computing majority by standard monotone circuits (gates of arity 2) we
provide an explicit monotone circuit of depth $O(\log_2^{5/3} n)$. The
construction is a combination of several known and not too complicated ideas.
  For arbitrary arity of gates $k$ we provide a new sorting network
architecture inspired by representation of inputs as a high-dimensional cube.
As a result we provide a simple construction that improves previous upper bound
of $4 \log_k^2 n$ to $2 \log_k^2 n$. We prove the similar bound for the depth
of the circuit computing majority of $n$ bits consisting of gates computing
majority of $k$ bits. Note, that for both problems there is an explicit
construction of depth $O(\log_k n)$ known, but the construction is complicated
and the constant hidden in $O$-notation is huge.
