Trace representation of some generalized cyclotomic sequences of length pq

This paper contributes to trace representation of some generalized cyclotomic sequences of length image which are defined by Ding and Helleseth. From the relations between these sequences and the Legendre sequence, we firstly confirm the defining pairs of these sequences of arbitrary order. Then, we obtain their trace representation, from which we give their linear complexity using Key’s method. It can be seen that Bai et al.’s conclusion is a special case of our result when the order is two. Finally, an example is given to illustrate the validity of our result.