  We build boolean circuits of size $O(nm^2)$ and depth $O(\log(n) + m
\log(m))$ for sorting $n$ integers each of $m$-bits. We build also circuits
that sort $n$ integers each of $m$-bits according to their first $k$ bits that
are of size $O(nmk(1 + \log^*(n) - \log^*(m)))$ and depth $O(\log^{3}(n))$.
This improves on the result of Asharov et al. arXiv:2010.09884 and resolves
some of their open questions.
