Comments on "A New Method to Compute the 2-Adic Complexity of Binary Sequences"

We show that there is a very simple approach to determine the 2-adic complexity of periodic binary sequences with ideal two-level autocorrelation. This is the first main result by H. Xiong, L. Qu, and C. Li, IEEE Transactions on Information Theory, vol. 60, no. 4, pp. 2399-2406, Apr. 2014, and the main result by T. Tian and W. Qi, IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 450-454, Jan. 2010. 1 A Very Simple Approach Let S = {si} +∞ i=0 be a periodic binary sequence with period N , and S(x) = ∑N−1 i=0 six i ∈ Z[x]. Let us write S(2) 2N − 1 = ∑N−1 i=0 si2 i 2N − 1 = p q with 0 ≤ p ≤ q, and gcd(p, q) = 1. Definition 1 ([3]) With the notations as above, the 2-adic complexity Φ(S) of S is the real number log2 q. Remark 1 If gcd(S(2), 2 −1) = 1, then the 2-adic complexity Φ(S) of S achieves the maximum value log2(2 N − 1). For any 0 ≤ τ < N , the autocorrelation of S at shift τ is defined by