Difference systems of sets (DSSs) introduced by Levenstein are combinatorial structures used to construct comma-free codes for synchronization. In this letter, two classes of optimal DSSs are presented. One class is obtained based on q-ary ideal sequences with d-form property and difference-balanced property. The other class of optimal and perfect DSSs is derived from perfect ternary sequences given by Ipatov in 1995. Compared with known constructions (Zhou, Tang, Optimal and perfect difference systems of sets from q-ary sequences with difference-balanced property, Des. Codes Cryptography, 57(2), 215-223, 2010), the proposed DSSs lead to comma-free codes with nonzero code rate.

In this paper, for any given prime power q, using Helleseth-Gong sequences with ideal auto-correlation property, we propose a class of perfect sequences of length (qm-1)/(q-1). As an application, a subclass of constructed perfect sequences is used to design optimal and perfect difference systems of sets.

A unified construction for transforming binary sequences of balance or unbalance into quaternary sequences is presented. On the one hand, when optimal and balanced binary sequences with even period are employed, our construction is exactly the same Jang, et al.'s and Chung, et al.'s ones, which result in balanced quaternary sequences with optimal autocorrelation magnitude. On the other hand, when ideal and balanced binary sequences with odd period N are made use of, our construction produces new balanced quaternary sequences with optimal autocorrelation value (OAV), in which there are N distinct sequences in terms of cyclic shift equivalence, and includes Tang, et al.'s and Jang, et al.'s ones as special cases. In addition, when binary sequences without period 2n-1 or balance are employed, the transformation of Jang, et al.'s method is invalid, however, the proposed construction works very good. As a consequence, this unified construction allows us to construct optimal and balanced quaternary sequences from ideal/optimal balanced binary sequences with arbitrary period.