Enumeration and criteria for cyclically shift-distinct GMW sequences

Gordon-Mills-Welch (GMW) sequences (also called cascaded GMW sequences) have two-level autocorrelations. This property makes them widely used in various communication and cryptographic systems. The generation of q-ary GMW sequences of period q^n-1 involves three types of parameters. To determine whether GMW sequences are cyclically shift-distinct for differing parameters has remained an open question until now. In this paper, we completely solve this problem for varying all three types of parameters. We find a criterion for cyclically shift-distinct q-ary GMW sequences of period q^n-1, and 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