Difference Systems of Sets with Size 2

Abstract

Difference systems of sets (DSS) are combinatorial structures introduced by Levenshtein in (Probl Peredachi Inform 7(3):215---222, 1971), which are a generalization of cyclic difference sets and arise in connection with code synchronization. In this paper, we consider a collection of pairs in a finite field of a prime order $$p=ef+1$$p=ef+1 to be a regular DSS with parameters $$(p,2,f,\rho )$$(p,2,f,?). We give a lower bound on the parameter $$\rho $$? using cyclotomic numbers for $$e=3$$e=3 and 4. In addition, we present a condition for which the collection of pairs forms an optimal DSS for $$e=4$$e=4.