Recursive bounds for perfect hashing

Let k⩽b be positive integers. A family C of sequences of length t over an alphabet of size b is called k-separated if for any k distinct members of C, there is a coordinate in which they mutually differ. Let N(t,b,k) denote the maximum size of such a family. This function has been studied extensively, mainly in the context of perfect hashing. Here we slightly improve a recent bound of Dyachkov, showing that for all t