For fixed integers $b\geq k$, the problem of perfect $(b,k)$-hashing asks for the asymptotic growth of largest subsets of $\{1,2,\ldots,b\}^n$ such that for any $k$ distinct elements in the set, there is a coordinate where they all differ. An important asymptotic upper bound for general $b, k$, was derived by Fredman and Koml\'os in the '80s and improved for certain $b\neq k$ by K\"orner and Marton and by Arikan. Only very recently better bounds were derived for the general $b,k$ case by Guruswami and Riazanov, while stronger results for small values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan and by Costa and Dalai. In this paper, we both show how some of the latter results extend to $b\neq k$ and further strengthen the bounds for some specific small values of $b$ and $k$. The method we use, which depends on the reduction of an optimization problem to a finite number of cases, shows that further results might be obtained by refined arguments at the expense of higher complexity.

翻译：对于固定整数 $b\ geq k$, 完美的美元( b, k) 问题要求最大子集的1, 2, doldots, b ⁇ n$的零增长幅度要小一些, 因此对于每组中任何1, 2,\ ldots, b ⁇ n$, 任何美元不同的元素, 都有不同的坐标。 对于普通的 b, k$, k美元, Fredman 和 Koml\' os, 在80年代得出了一个重要的零花钱上限。 K\\\\ neq k$, K\ orner 和 Marton 以及 Arikan 改进了某些 $ ( $) 。 直到最近, Guruswami 和 Riazanov 对普通的 $b, k$ ( $ k$) 的情况才得出了更好的界限, 而 Arikan、 Dalai、 Guruswami 和 Radhakrishishnan 以及 Costa 和 Dalai Darai 。 在本文中, 中, 我们都展示了其中某些后项结果是如何将某些小数额降低 和美元 和 美元 美元 。