Note on a construction of short k-radius sequences

A sequence over an n-element alphabet is called a k-radius sequence if any two distinct elements of the alphabet occur within distance k of each other somewhere in the sequence. Let fk(n) be the shortest length of a k-radius sequence over an n-element alphabet. In this note we give a new construction of “short” k-radius sequences, based on the concept of a difference family. It allows us to prove that for every fixed positive integer k there are infinitely many values of n such that fk(n)=1kn2+O(n). This way we improve an earlier result by Blackburn and McKee (2012) who showed that the same formula holds for infinitely many values of n only when k satisfies some special conditions.