改进的$(b,k)$哈希边界
Improved Bounds for $(b,k)$-hashing
摘要：
对于固定整数$b\geq k$，计算机科学和组合学中的一个相关兴趣问题是，用$n$确定最大集合的渐近增长，对于该最大集合，存在$n$函数的$(b, k)$ -散列族。等价地，确定$\{1,2,\ldots,b\}^n$的最大子集的渐近增长，使得对于集合中的任何$k$不同元素，存在它们都不同的坐标。一般$b, k$的一个重要的渐近上界是由Fredman和Komlós在80年代推导出来的，并由Körner和Marton以及Arikan对某些$b\neq k$进行了改进。只是最近更好的界限推导出一般$b,k$的情况下，由Guruswami和Riazanov，而更强的结果，小值的$b=k$，由Arikan，达赖，Guruswami和Radhakrishnan和科斯塔和达赖。在本文中，我们都显示了一些后者的结果扩展到$b\neq k$，并进一步加强了一些特定的小值的$b$和$k$的界限。我们使用的方法，这取决于减少的优化问题的有限数量的情况下，表明，进一步的结果可能会得到细化的参数，以牺牲更高的复杂性，可以通过使用更复杂和优化的算法方法来减少。
英文摘要：
For fixed integers $b\geq k$, a problem of relevant interest in computer science and combinatorics is that of determining the asymptotic growth, with $n$, of the largest set for which a $(b, k)$-hash family of $n$ functions exists. Equivalently, determining the asymptotic growth of a largest subset 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 Komlós in the '80s and improved for certain $b\neq k$ by Körner 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 which could be reduced by using more sophisticated and optimized algorithmic approaches.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
