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Register a new userThe 2-Adic Complexity of Two Classes of Binary Sequences with Interleaved Structure
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Minghui Yang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The autocorrelation values of two classes of binary sequences are shown to be good in [6]. We study the 2-adic complexity of these sequences. Our results show that the 2-adic complexity of such sequences is large enough to resist the attack of the rational approximation algorithm.
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The generalized binary sequences of order 2 have been used to construct good binary cyclic codes [4]. The linear complexity of these sequences has been computed in [2]. The autocorrelation values of such sequences have been determined in [1] and [3]. Some lower bounds of 2-adic complexity for such sequences have been presented in [5] and [7]. In this paper we determine the exact value of 2-adic complexity for such sequences. Particularly, we improve the lower bounds presented in [5] and [7] and the condition for the 2-adic complexity reaching the maximum value.
We determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences by using cyclotomic numbers of order four, Gauss periods and quadratic Gauss sum on finite field $mathbb{F}_q$ and valued in $mathbb{Z}_{2^N-1}$ where $q equiv 5pmod 8$ is a prime number and $N=2q$ is the period of the DHM sequences.
Binary periodic sequences with good autocorrelation property have many applications in many aspects of communication. In past decades many series of such binary sequences have been constructed. In the application of cryptography, such binary sequences are required to have larger linear complexity. Tang and Ding cite{X. Tang} presented a method to construct a series of binary sequences with period 4$n$ having optimal autocorrelation. Such sequences are interleaved by two arbitrary binary sequences with period $nequiv 3pmod 4$ and ideal autocorrelation. In this paper we present a general formula on the linear complexity of such interleaved sequences. Particularly, we show that the linear complexity of such sequences with period 4$n$ is not bigger than $2n+2$. Interleaving by several types of known binary sequences with ideal autocorrelation ($m$-sequences, Legendre, twin-prime and Halls sequences), we present many series of such sequences having the maximum value $2n+2$ of linear complexity which gives an answer of a problem raised by N. Li and X. Tang cite{N. Li}. Finally, in the conclusion section we show that it can be seen easily that the 2-adic complexity of all such interleaved sequences reaches the maximum value $log_{2}(2^{4n}-1)$.
Via interleaving Ding-Helleseth-Lam sequences, a class of binary sequences of period $4p$ with optimal autocorrelation magnitude was constructed in cite{W. Su}. Later, Fan showed that the linear complexity of this class of sequences is quite good cite{C. Fan}. Recently, Sun et al. determined the upper and lower bounds of the 2-adic complexity of such sequences cite{Y. Sun3}. We determine the exact value of the 2-adic complexity of this class of sequences. The results show that the 2-adic complexity of this class of binary sequences is close to the maximum.
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 $n$.
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