New upper bounds for $(b,k)$-hashing

For fixed integers $b\ge k$, the problem of perfect $(b,k)$-hashing asks for the asymptotic growth of largest subsets of $\{1,2,\dots ,b{\}}^n$ 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 $b\mathrm{\ne }k$ 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 $b\neq k$andfurtherstrengthentheboundsforsomespecificsmallvaluesof$b$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.