New upper bounds for (b, k)-hashing

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, was derived by Fredman and Komlós in the ‘80s and improved for certain by Körner and Marton and by Arikan. Only very recently better bounds were derived for the general case by Guruswami and Riazanov, while stronger results for small values of 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 and further strengthen the bounds for some specific small values of and. 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.