击败Fredman-Komlós获得完美的$k$散列
Beating Fredman-Komlós for perfect $k$-hashing
摘要：
我们说子集$C \subseteq \{1,2,\dots,k\}^n$是$k$散列码（也称为$k$分离的），如果对于来自$C$的每个$k$码字子集，存在一个坐标，其中所有这些码字具有不同的值。理解$k$ -散列码的最大可能速率（以比特为单位）（定义为$(\log_2 |C|)/n$）是一个经典问题。它出现在两个等价的上下文中：（i）将$N$元素的全域映射到$\{1,2,\dots,k\}$中的完美散列族的可能的最小大小，以及（ii）对于特定组合信道，利用大小小于$k$的列表进行解码的零错误容量。对于任何$k \geq 4$，Fredman和Komlós在1984年得到了一个关于$k$ -散列码的速率的$k!/k^{k-1}$的一般上界（在大$n$的限制下）。虽然已经获得了$k=4$的更好的界限，但它们的原始界限仍然是每个$k \ge 5$的最佳界限。在这项工作中，我们得到了第一个改进的Fredman-Komlós界的每$k \ge 5$。虽然我们得到明确的（数值）界为$k=5,6$，对于较大的$k$，我们只表明，FK界可以提高一个积极的，但未指定的，量。在单纯形上某个多项式优化问题的最优值的猜想下，我们的方法允许对每个$k$计算有效界。
英文摘要：
We say a subset $C \subseteq \{1,2,\dots,k\}^n$ is a $k$-hash code (also called $k$-separated) if for every subset of $k$ codewords from $C$, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as $(\log_2 |C|)/n$, of a $k$-hash code is a classical problem. It arises in two equivalent contexts: (i) the smallest size possible for a perfect hash family that maps a universe of $N$ elements into $\{1,2,\dots,k\}$, and (ii) the zero-error capacity for decoding with lists of size less than $k$ for a certain combinatorial channel. A general upper bound of $k!/k^{k-1}$ on the rate of a $k$-hash code (in the limit of large $n$) was obtained by Fredman and Komlós in 1984 for any $k \geq 4$. While better bounds have been obtained for $k=4$, their original bound has remained the best known for each $k \ge 5$. In this work, we obtain the first improvement to the Fredman-Komlós bound for every $k \ge 5$. While we get explicit (numerical) bounds for $k=5,6$, for larger $k$ we only show that the FK bound can be improved by a positive, but unspecified, amount. Under a conjecture on the optimum value of a certain polynomial optimization problem over the simplex, our methods allow an effective bound to be computed for every $k$.
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。
