On the 2-Adic Complexity and the k-Error 2-Adic Complexity of Periodic Binary Sequences

In this paper, we point out a significant difference between the linear complexity and the 2-adic complexity of periodic binary sequences. The concept of the symmetric 2-adic complexity of periodic binary sequences is presented based on this observation. We determine the expected value of the 2-adic complexity and derive a lower bound on the expected value of the symmetric 2-adic complexity of periodic binary sequences. Because the 2-adic complexity of periodic binary sequences is unstable, we present the concepts of the k-error 2-adic complexity and the k-error symmetric 2-adic complexity, and lower bounds on them are also derived.