Polyphase power-residue sequences

Polyphase versions of the well-known binary Legendre or quadratic-residue sequences were first described and analysed by Sidelnikov over thirty years ago, but have since received very little attention. Here, it is shown that these q-phase sequences of prime length L can also be constructed from the index sequence of length L or, equivalently, from the cosets of qth power residues and non-residues. These sequences are also shown to fall into two classes, each of which has well-defined periodic autocorrelation functions and merit factors. Class-I sequences exist for prime L ≡ q + 1 mod 2q and q even, whereas class-II sequences correspond to primes of the form L ≡ 1 mod 2q and are available for all values of q. Class-I sequences have complex out-of-phase correlation values with magnitudes less than or equal to √5, whereas for class-II sequences these are purely real with magnitude less than or equal to 3. Some other properties are also investigated.