A lower bound on the 2-adic complexity of Ding-Helleseth generalized cyclotomic sequences of period pnp^n

Let $p$ be an odd prime, $n$ a positive integer and $g$ a primitive root of $p^n$. Suppose $D_i^{(p^n)}=\{g^{2s+i}|s=0,1,2,\cdots,\frac{(p-1)p^{n-1}}{2}\}$, $i=0,1$, is the generalized cyclotomic classes with $Z_{p^n}^{\ast}=D_0\cup D_1$. In this paper, we prove that Gauss periods based on $D_0$ and $D_1$ are both equal to 0 for $n\geq2$. As an application, we determine a lower bound on the 2-adic complexity of a class of Ding-Helleseth generalized cyclotomic sequences of period $p^n$. The result shows that the 2-adic complexity is at least $p^n-p^{n-1}-1$, which is larger than $\frac{N+1}{2}$, where $N=p^n$ is the period of the sequence.Comment: 1

Linear complexity of generalized cyclotomic sequences of order 4 over F_l

Generalized cyclotomic sequences of period pq have several desirable randomness properties if the two primes p and q are chosen properly. In particular,Ding deduced the exact formulas for the autocorrelation and the linear complexity of these sequences of order 2. In this paper, we consider the generalized sequences of order 4. Under certain conditions, the linear complexity of these sequences of order 4 is developed over a finite field F_l. Results show that in many cases they have high linear complexity.Comment: Since there is a crucial error in Theorem 1 in the first version, we replace it by the new on