  For fixed integers $b\geq k$, the problem of perfect $(b,k)$-hashing asks for
the asymptotic growth of largest subsets of $\{1,2,\ldots,b\}^n$ 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 $b, k$, was derived by
Fredman and Koml\'os in the '80s and improved for certain $b\neq k$ by K\"orner
and Marton and by Arikan. Only very recently better bounds were derived for the
general $b,k$ case by Guruswami and Riazanov, while stronger results for small
values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan
and by Costa and Dalai.
  In this paper, we both show how some of the latter results extend to $b\neq
k$ and further strengthen the bounds for some specific small values of $b$ and
$k$. 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.
