Determination of the Autocorrelation Distribution and 2-Adic Complexity of Generalized Cyclotomic Binary Sequences of Order 2 with Period pq

The generalized cyclotomic binary sequences $S=S(a, b, c)$ with period $n=pq$ have good autocorrelation property where $(a, b, c)in {0, 1}^3$ and $p, q$ are distinct odd primes. For some cases, the sequences $S$ have ideal or optimal autocorrelation. In this paper we determine the autocorrelation distribution and 2-adic complexity of the sequences $S=S(a, b, c)$ for all $(a, b, c)in {0, 1}^3$ in a unified way by using group ring language and a version of quadratic Gauss sums valued in group ring $R=mathbb{Z}[Gamma]$ where $Gamma$ is a cyclic group of order $n$.