Sorting Short Integers

We build boolean circuits of size $O(nm^2)$ and depth $O(\log(n) + m \log(m))$ for sorting $m$-bit itnegers. We build also circuits that sort $m$-bit integers 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.