Computational Results for Regular Difference Systems of Sets Attaining or Being Close to the Levenshtein Bound

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 = e f + 1 to be a regular DSS for an integer m , and give a lower bound on the parameter ρ 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 ρ is independent of the choice of e − 1 subsets. In addition, we present some computational results for DSSs with block sizes f l o o r ( e / 2 ) and f l o o r ( e / 3 ) , whose parameter ρ attains or comes close to the Levenshtein bound for p < 100 .