Linear complexity of polyphase power residue sequences

The well known family of binary Legendre or quadratic residue sequences can be generalised to the multiple-valued case by employing a polyphase representation. These p-phase sequences, with p prime, also have prime length L, and can be constructed from the index sequence of length L or, equivalently, from the cosets of pth power residues and non-residues modulo-L. The linear complexity of these polyphase sequences is derived and shown to fall into four classes depending on the value assigned to b(0), the initial digit of the sequence, and on whether p belongs to the set of pth power residues or not. The characteristic polynomials of the linear feedback shift registers that generate these sequences are also derived.