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

Abstract

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 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 $p$ . To investigate some relation between Fermat-quotient sequences and Frank-Zadoff sequences and to construct optimal families including these sequences, we introduce generators of $p$ -ary polyphase sequences of period $p^{2}$ using their $p\times p$ array structures. We call an optimal generator to be the generator of some $p$ -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 $p-1$ 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 $p\geq 11$ . The relation between the Fermat-quotient sequences and the Frank-Zadoff sequences is determined as a by-product.