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

TL;DR

This paper analyzes the autocorrelation distribution and 2-adic complexity of generalized cyclotomic binary sequences of order 2 with period pq, providing a unified approach using group rings and quadratic Gauss sums.

Contribution

It determines the autocorrelation distribution and 2-adic complexity for all parameter cases of these sequences using a novel algebraic framework.

Findings

Sequences can have ideal or optimal autocorrelation properties.

Explicit autocorrelation distributions are derived for all parameter cases.

The 2-adic complexity is characterized for the sequences.

Abstract

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 $n$.