Constructing k-radius sequences

An n-ary k-radius sequence is a finite sequence of elements taken from an alphabet of size n such that any two distinct elements of the alphabet occur within distance k of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc to model a caching strategy for computing certain functions on large data sets such as medical images. Let fk(n) be the shortest length of any k-radius sequence. We improve on earlier estimates for fk(n) by using tilings and logarithms. The main result is that fk(n) ∼ 1 k (n 2 ) as n → ∞ whenever there exists a tiling of Zπ(k) by a certain cluster of k hypercubes. In particular this result holds for infinitely many k, including all k ≤ 194 and all k such that k + 1 or 2k + 1 is prime. For certain k, in particular when 2k + 1 is prime, we get a sharper error term using the theory of logarithms.