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 (9h
₁
²
+ 3h
₂
²
) for (h
₁
, h
₂
) ∈ Z
²
and we also give the exact number of occurrences of each element from α
₃
in the code. Finally, all possible lengths (≤100) of PRUCs over α
₅
and α
₇
together with their phase distributions are provided.