New Perfect Sequences and Uncorrelated Optimal ZCZ Sequences over PSK+ Alphabet of Small Size from Cyclic Relative Difference Sets

In this paper, we present a new construction framework for perfect sequences and uncorrelated optimal zero-correlation zone (ZCZ) sequence families over PSK+ alphabets, using the algebraic structure of cyclic relative difference sets (RDSs). By introducing the notion of characteristic sequences of an RDS, we embed constituent PSK seed sequences-such as those from uncorrelated optimal ZCZ families-into structured supports determined by the RDS. This approach enables the construction of sequences whose lengths or ZCZ widths are integer multiples (not necessarily perfect squares) of q l −1 q−1 for prime power q and some integer l. Our constructions include cases where smaller phase sizes are achieved for the same sequence parameters and the same peak-to-average power ratio (PAPR). This work also appears to be the first to employ cyclic RDSs in the construction of perfect sequences or uncorrelated optimal ZCZ sequence families.