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 $k\ge 4$. 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 present a method to obtain the first improvement to the Fredman-Komlós bound for every $k\ge 5$.

我们说一个子集 $C\subseteq {\{1,2,\dots ,ķ\}}^n$如果对于来自C的k个码字的每个子集，都存在一个坐标，其中所有这些码字具有不同的值，则它是一个k哈希码（也称为k分隔）。了解最大可能速率（以位为单位），定义为$({\mathrm{日志}}_2|C|)/n$, k哈希码是一个经典问题。它出现在两个等效的上下文中：(i) 将包含N个元素的全域映射到的完美散列族的最小尺寸$\{1,2,\dots ,ķ\}$，以及（ii）对于某个组合通道，使用大小小于k的列表进行解码的零错误容量。

的一般上限 $ķ！/{ķ}^{ķ-1}$Fredman和Komlós在1984年针对任何$ķ\ge 4$. 虽然已经获得了更好的界限$ķ=4$，他们原来的界限仍然是最知名的 $ķ\ge 5$. 在这项工作中，我们提出了一种方法来获得对每个 Fredman-Komlós 界的第一次改进$ķ\ge 5$.