The Linear Complexity of Whiteman's Generalized Cyclotomic Sequences of Period $p^{m+1}q^{n+1}$

In this paper, we mainly get three results. First, let p, q be distinct primes with ((p-1)p,(q-1)q)=(p-1,q-1)=e ; we give a method to compute the linear complexity of Whiteman's generalized cyclotomic sequences of period p^m+1qⁿ⁺¹. Second, if e=4, we compute the exact linear complexity of Whiteman's generalized cyclotomic sequences. Third, if p≡q 5 (mod 8), gcd(p-1, q-1)=4, and we fix a common primitive root g of both p and q, then 2∈H₀=(g), which is a subgroup of the multiplicative group Z*_pq, if and only if Whiteman's generalized cyclotomic numbers of order 4 depend on the decomposition pq=a²+4b² with 4|b.