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 Koml\'{o}s 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-Koml\'{o}s 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 $k$.
[1]
J. Radhakrishnan.
Entropy and Counting ∗
,
2001
.
[2]
János Körner,et al.
New Bounds for Perfect Hashing via Information Theory
,
1988,
Eur. J. Comb..
[3]
A. Nilli.
Perfect Hashing and Probability
,
1994,
Combinatorics, Probability and Computing.
[4]
Erdal Arikan,et al.
An upper bound on the zero-error list-coding capacity
,
1994,
IEEE Trans. Inf. Theory.
[5]
Venkatesan Guruswami,et al.
An improved bound on the zero-error list-decoding capacity of the 4/3 channel
,
2017,
2017 IEEE International Symposium on Information Theory (ISIT).
[6]
Efe A. Ok.
Real analysis with economic applications
,
2007
.
[7]
J. Komlos,et al.
On the Size of Separating Systems and Families of Perfect Hash Functions
,
1984
.
[8]
Erdal Arikan.
A Bound on the Zero-Error List Coding Capacity
,
1993,
Proceedings. IEEE International Symposium on Information Theory.
[9]
Peter Elias,et al.
Zero error capacity under list decoding
,
1988,
IEEE Trans. Inf. Theory.