Linear complexity and correlation of a class of binary cyclotomic sequences

Let p₁,p₂,....,p_n p 1, p 2, ..., p n be distinct odd primes and let e₁,e₂,....,e_n e 1, e 2, ..., e n be positive integers. Based on cyclotomic classes proposed by Ding and Helleseth (Finite Fields Appl 4:140-166, 1998), a binary cyclotomic sequence of period p₁^e₁p₂ ^e₂... p_n^e_n p 1 e 1 p 2 e 2 ... p n e n is defined and denoted by s s Υ. The linear complexity of s s Υ is determined and is proved to be greater than or equal to (p ₁^e₁p₂^e₂... p_n^e_n-1)/2 (p 1 e 1 p 2 e 2 ... p n e n - 1) / 2. The autocorrelation function of s s Υ is explicitly computed. Let l \in \1,2,....,n\ l 1, 2, ..., n . We also explicitly compute the crosscorrelation function of s s Υ and the Legendre sequence L_pl L p l with respect to p-l p l. It is shown that s s Υ and L_pl L p l have two-level or three-level crosscorrelation, and all their two-level crosscorrelation functions are determined.