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.

翻译：我们建造了大小为$O(nm%2) $和深度为$O(log(n) + m\log(m) ) $(m) 的布林电路,用于分拣美元- 位相。我们建造的电路也按照大小为$(nmk(1 +\log* * (n) -\log* (m))) $和深度为$O(\log(n) + m\log(m) ) $(m) 美元) 和深度为$O(g(n) + m\log(m) ) 美元。根据Asharov 等人(arxiv:2010.09884) 的结果,这些电路可以将美元- 位整数排序为$美元,并解决其中一些开放的问题。