Recursive bounds for perfect hashing

Abstract 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 k ⩽ b , N ( t , b , k )⩽ tb −( k −1)( t −1). This implies that if k ⩽ b and t is divisible by k −1, then N ( t , b , k )⩽( k −1) b t /( k −1) −( k −1) 2 .