Further strengthening of upper bounds for perfect $k$-Hashing

For a fixed integer $k$, 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 perfect $k$-hash family of $n$ functions exists. Equivalently, determining the asymptotic growth of a largest subset of $\{1,2,\ldots,k\}^n$ 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 $k$ was derived by Fredman and Koml\'os in the '80s. Only very recently this was improved for general $k$ by Guruswami and Riazanov while stronger results for small values of $k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan and by Dalai and Costa. In this paper, we further improve the bounds for $5\leq k \leq 8$. 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.