On Prime Root-of-Unity Sequences With Perfect Periodic Correlation

In this paper, perfect root-of-unity sequences (PRUS) with entries in α_p = {x ∈ BBC | x^p = 1} (where p is a prime) are studied. A lower bound on the number of distinct phases that are used in PRUS over α_p is derived. We show that PRUS of length L ≥ p(p-1) must use all phases in α_p. Certain conditions on the lengths of PRUS are derived. Showing that the phase values of PRUS must follow a given difference multiset property, we derive a set of equations (which we call the principal equations) that give possible lengths of a PRUS over α_p together with their phase distributions. The usefulness of the principal equations is discussed, and guidelines for efficient construction of PRUS are provided. Through numerical results, contributions also are made to the current state-of-knowledge regarding the existence of PRUS. In particular, a combination of the developed ideas allowed us to numerically settle the problem of existence of PRUS with (L, p)=(28, 7) within about two weeks-a problem whose solution (without using the ideas in this paper) would likely take more than three million years on a standard PC.