A new unified construction of perfect root-of-unity sequences

Perfect root-of-unity sequences (PRUS) have found applications in numerous spread spectrum systems such as pulse compression radars, DS/SSMA, FH/SSMA, etc. A unified PRUS construction, which includes all known constructions as special cases, was previously devised by this author. In this paper, a class of root-of-unity sequence sets with certain interesting correlation properties is identified as a very important source of PRUS. By deriving a general construction of such sequence sets, a new, even more general unified PRUS construction is obtained. In addition, a new lower bound on the total number of PRUS with given alphabet size and sequence length is derived by determining the exact number of PRUS obtainable from the construction. The bound is tight for all cases, in which the exact values are known. Furthermore, it is proved that there is no new PRUS obtainable through the application of various perfectness-invariant transformations and the direct product construction. This gives strong evidences to support the conjecture that the new unified construction in fact describes all PRUS that exist.