Difference systems of sets (DSSs) are combinatorial structures arising in connection with code synchronization that were introduced by Levenshtein in 1971, and are a generalization of cyclic difference sets. In this paper, we consider a collection of m-subsets in a finite field of prime order p = ef + 1 to be a regular DSS for an integer m, and give a lower bound on the parameter. of the DSS using cyclotomic numbers. We show that when we choose (e -1)-subsets from the multiplicative group of order e, the lower bound on. is independent of the choice of e - 1 subsets. In addition, we present some computational results for DSSs with block sizes f loor(e/2) and f loor(e/3), whose parameter. attains or comes close to the Levenshtein bound for p < 100. (C) 2016 Wiley Periodicals, Inc.