Families of perfect polyphase sequences from the array structure of Fermat-Quotient sequences and Frank-Zadoff sequences

We show that a p-ary polyphase sequence of period p
²
from the Fermat quotients is `perfect.' That is, its periodic autocorrelation is zero for all non-trivial shifts. We call this Fermat-Quotient sequences. Using this and the fact that the Frank-Zadoff sequences (which is known to be also perfect), we propose a collection of `optimum' families of perfect polyphase sequences in the sense of Sarwate Bound. That is, the cross-correlation of two members in a family is upper bounded by p. We may say these families are `completely optimum' since the cross-correlation of any two members in a family is exactly p for all phase-shifts.