Perfect Root-Of-Unity Codes with prime-size alphabet
Mojtaba Soltanalian, Petre Stoica · 2011
In this paper, Perfect Root-of-Unity Codes (PRUCs) with entries in αp= {x ∈ C | xp= 1} where p is a prime are studied. A lower bound on the number of distinct phases in PRUCs over αpis derived. We show that PRUCs of length L ≥ p(p - 1) must use all phases in αp. It is also shown that if there exists a PRUC of length L over αpthen p divides L. We derive equations (which we call principal equations) that give possible lengths of a PRUC over αptogether 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) ∈ Z2and we also give the exact number of occurrences of each element from α3in the code. Finally, all possible lengths (≤100) of PRUCs over α5and α7together with their phase distributions are provided.