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

Abstract

Abstract. In this paper, a new method is presented to compute the 2-adic complexity of pseudo-random sequences. With this method, the 2-adic complexities of all the known sequences with ideal 2-level autocorrelation are uniformly determined. Results show that their 2-adic complexities equal their periods. In other words, their 2-adic complexities attain the maximum.Moreover, 2-adic complexities of two classes of optimal autocorrelation sequences with period N ≡ 1 mod 4, namely Legendre sequences and Ding-Helleseth-Lam sequences, are investigated.Besides, this method also can be used to compute the linear complexity of binary sequences regarded as sequences over other finite fields.

A lower bound on the 2-adic complexity of the modified Jacobi sequence

“…In this section, using the periods which have been determined in Section 3, we will give a lower bound on the 2-adic complexity of modified Jacobi sequence. In order to derive the lower bound, the following two results will be useful, which can be found in [18] and [3] respectively. Lemma 4 [18] Viewing A as a matrix over the rational fields Q, if det(A) = 0, then gcd S(2), 2 N − 1 |gcd det(A), 2 N − 1 .…”

“…In 2010, Tian and Qi showed that the 2-adic complexity of all the binary m-sequences is maximal [16]. Afterwards, Xiong et al [18] presented a new method using circulant matrices to compute the 2-adic complexities of binary sequences. They showed that all the known sequences with ideal 2-level autocorrelation have maximum 2-adic complexity.…”

“…Moreover, they also proved that the 2-adic complexities of Legendre sequences and Ding-Helleseth-Lam sequences with optimal autocorrelation are also maximal. Then, using the same method as that in [18], Xiong et al [19] pointed out that two other classes of sequences based on interleaved structure have also maximal 2-adic complexity. One of these two classes of sequences was constructed by Tang and Ding [14], which has optimal autocorrelation, the other was constructed by Zhou et al [24], which is optimal with respect to the Tang-Fan-Matsufuji bound [15].…”

“…Green and Choi have proved that these sequences have large linear complexity and low autocorrelation in many cases. But, as far as the authors known, among Jacobi sequence family, there is no other result about the 2-adic complexities of these sequences other than twinprime sequences, which has ideal autocorrelation and has been proved to be with maximal 2-adic complexity by Xiong et al [18], and another class of sequences constructed from Whiteman generalized cyclotomic classes of order 2 has been proved to be also with maximal 2-adic complexity by Zeng et al [22].…”

The exact autocorrelation distribution and 2-adic complexity of a class of binary sequences with almost optimal autocorrelation

“…Moreover, in [2], Ding et al proved that the 2-adic complexities of Legendre sequences and Ding-Helleseth-Lam sequences with optimal autocorrelation are also maximal. Then, using the same method as that in [22], Xiong et al [23] pointed out that two other classes of sequences based on interleaved structure have also maximal 2-adic complexity. One of these two classes of sequences was constructed by Tang and Ding [15], which has optimal autocorrelation, the other was constructed by Zhou et al [24], which is optimal with respect to the Tang-Fan-Matsufuji bound [16].…”