Perfect Root-Of-Unity Codes with prime-size alphabet

In this paper, Perfect Root-of-Unity Codes (PRUCs) with entries in α_p = {x ∈ C | x^p = 1} where p is a prime are studied. A lower bound on the number of distinct phases in PRUCs over α_p is derived. We show that PRUCs of length L ≥ p(p - 1) must use all phases in α_p. It is also shown that if there exists a PRUC of length L over α_p then p divides L. We derive equations (which we call principal equations) that give possible lengths of a PRUC over α_p together with their phase distribution. Using these equations, we prove for example that the length of a 3-phase perfect code must be of the form L = 1/4 (9h₁² + 3h₂²) for (h₁, h₂) ∈ Z² and we also give the exact number of occurrences of each element from α₃ in the code. Finally, all possible lengths (≤100) of PRUCs over α₅ and α₇ together with their phase distributions are provided.