Binary Sequences Derived from Dickson Permutation Polynomials over Binary Extension Field

In this paper, based on a result of Lidl and Mullen (Mathematical Journal of Okayama University, 1991), the maximum length and the second maximum length that can be attained by cycles of Dickson permutation polynomial (of the first kind) with parameter 1 are studied. Necessary and sufficient conditions for these two lengths to be attained are given, which are connected with Fermat primes and Mersenne primes, respectively. Furthermore, a class of coordinate sequences that maintains a large period is obtained, which is shown to be the coordinate sequences derived from cycles of the second maximum length. Explicit formulas for their periodicity and shift-equivalences are also presented.