Enumeration and criteria for cyclically shift-distinct GMW sequences

Gordon-Mills-Welch (GMW) sequences (also called cascaded GMW sequences) hav e two-level autocorrelations, This property makes them widely used in vario us communication and cryptographic systems. The generation of q-ary GMW seq uences of period q(n) - 1 involves three types of parameters. To determine whether GMW sequence are cyclically shift-distinct for differing parameters has remained an open question until now. In this paper, we completely solv e this problem for varying all three types of parameters. We find a criteri on for cyclically shift-distinct q-ary GMW sequences of period q(n) - 1, an d obtain the number of such sequences. For the special case of q = 2, this solution facilitates counting the number of cyclic Hadamard difference sets which correspond to binary GMW sequences of period 2(n) - 1.