A nonempty circular string C(x) of length n is said to be covered by a set U_k of strings each of fixed length k <= n iff every position in C(x) lies within an occurrence of some string u in U_k. In this paper we consider the problem of determining the minimum cardinality of a set U_k which guarantees that every circular string C(x) of length n >= k can be covered. In particular, we show how, for any positive integer m, to choose the elements of U_k so that, for sufficiently large k, u_k \approx sigma^(k-m), where u_k = |U_k| and sigma is the size of the alphabet on which the strings are defined. The problem has application to DNA sequencing by hybridization using oligonucleotide probes.