An -ary -radius sequence is a finite sequence of elements taken from an alphabet of size such that any two distinct elements of the alphabet occur within distance 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 be the shortest length of any -radius sequence. We improve on earlier estimates for by using tilings and logarithms. The main result is that as whenever there exists a tiling of by a certain cluster of hypercubes. In particular this result holds for infinitely many , including all and all such that or is prime. For certain , in particular when is prime, we get a sharper error term using the theory of logarithms. References