  Let $C\subseteq \{1,\ldots,k\}^n$ be such that for any $k$ distinct elements
of $C$ there exists a coordinate where they all differ simultaneously. Fredman
and Koml\'os studied upper and lower bounds on the largest cardinality of such
a set $C$, in particular proving that as $n\to\infty$, $|C|\leq \exp(n
k!/k^{k-1}+o(n))$. Improvements over this result where first derived by
different authors for $k=4$. More recently, Guruswami and Riazanov showed that
the coefficient $k!/k^{k-1}$ is certainly not tight for any $k>3$, although
they could only determine explicit improvements for $k=5,6$. For larger $k$,
their method gives numerical values modulo a conjecture on the maxima of
certain polynomials.
  In this paper, we first prove their conjecture, completing the explicit
computation of an improvement over the Fredman-Koml\'os bound for any $k$.
Then, we develop a different method which gives substantial improvements for
$k=5,6$.
