Optimal Families of Perfect Polyphase Sequences From the Array Structure of Fermat-Quotient Sequences

We show that a <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>-ary polyphase sequence of period <inline-formula> <tex-math notation="LaTeX">$p^{2}$ </tex-math></inline-formula> from the Fermat quotients is perfect. That is, its periodic autocorrelation is zero for all non-trivial phase shifts. We call this Fermat-quotient sequence. We propose a collection of optimal families of perfect polyphase sequences using the Fermat-quotient sequences in the sense of the Sarwate bound. That is, the cross correlation of two members in a family is upper bounded by <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>. To investigate some relation between Fermat-quotient sequences and Frank-Zadoff sequences and to construct optimal families including these sequences, we introduce generators of <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>-ary polyphase sequences of period <inline-formula> <tex-math notation="LaTeX">$p^{2}$ </tex-math></inline-formula> using their <inline-formula> <tex-math notation="LaTeX">$p\times p$ </tex-math></inline-formula> array structures. We call an optimal generator to be the generator of some <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>-ary polyphase sequences which are perfect and which gives an optimal family by the proposed construction. Finally, we propose an algebraic construction for optimal generators as another main result. A lot of optimal families of size <inline-formula> <tex-math notation="LaTeX">$p-1$ </tex-math></inline-formula> can be constructed from these optimal generators, some of which are known to be from the Fermat-quotient sequences or from the Frank-Zadoff sequences, but some families are new for <inline-formula> <tex-math notation="LaTeX">$p\geq 11$ </tex-math></inline-formula>. The relation between the Fermat-quotient sequences and the Frank-Zadoff sequences is determined as a by-product.