Gold Sequences

In this paper, for odd n and any k with gcd(n, k) = 1, new binary sequence families Sk of period 2n-1 are constructed. These families have maximum correlation $1+2^{n+3\\over 2}$, family size 22n+2n+1 and maximum linear span $n(n+1)\\over 2$. The correlation distribution of Sk is completely determined as well. Compared with the modified Gold codes with the same family size, the proposed families have the same period and correlation properties, but larger linear span. As good candidates with low correlation and large family size, the new families contain the Gold sequences and the Gold-like sequences. Furthermore, Sk includes a subfamily $\\mathcal{S}^k_1$ which has the same period, correlation distribution, family size and linear span as the family So(2) recently constructed by Yu and Gong. In particular, when k=1, $\\mathcal{S}^k_1$ is exactly So(2).

In this paper, a new family of binary sequences of period 2n-1 with low correlation is proposed for integer n = em and even m. The new family has family size 2n + 1 and maximum nontrivial correlation 2n+e/2 + 1 and 2n+e-1/2 + 1 for even and odd e respectively. Especially, for n = 2m and 3m, we obtain a new family of binary sequences with maximum nontrivial correlation 2n/2 + 1 + 1, and the obtained family is one of the binary families with best correlation among the known families with family size no less than their period 2n-1 for even n. Moreover, the correlation distribution of the new family is also determined.

The generalized binary bent sequences and bent-lifted binary sequences with optimal correlation and balance properties are introduced in [J.S. No, et al., (1997)]. In this paper, it is shown that new extended binary bent sequences can be derived from the bent-lifted binary sequences by applying the similar method used to find extended binary sequences from No sequences.

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated.

A new family of binary sequences with low 4-value autocorrelation of length 4p based on cyclotomic classes of order four is proposed, as well as the linear complexity and 2-adic complexity of these sequences are determined. The results show that the linear complexity of s is LC(s) = 4p−1, and the 2-adic complexity of s is Φ2s = log2 ((24p−1)/3), both of which are greater than half of the period. The sequences are safe enough to resist Berlekamp–Massey algorithm attacks and rational approximation algorithm attacks, which indicate that they are good sequences in communication.

In this paper we give a new family of elliptic curve pseudorandom binary sequences, and study the well-distribution, correlation, linear complexity, collision and avalanche effect of the family of sequences, by using the estimate of exponential sums over elliptic curves.

A real-valued bent function is newly defined and shown to be useful to design polyphase sequences with optimal correlation properties. As an application of practical importance, balanced quadriphase sequences with optimal correlation properties are designed and shown to be the sum of two binary {1,−1} bent sequences.

A method for the synthesis of an infinite family of binary sequences with perfectly flat cross-correlation sidelobes is presented. The corresponding postprocessing arrays are similarly two-valued, and so the sequences may be particularly suited to applications in high noise environments.

On The Linear Span of A Binary Sequence Family with Optimal Correlation Properties

In this paper, we construct two new families S and U of optimal quadriphase sequences. Compared to the family A constructed by Boztas et al and family D investigated by Tang et al respectively, the proposed families have the same optimal correlation properties and family size, but larger linear spans.

A binary sequence family ${\mathcal S}$ of length $n$ and size $M$ can be characterized by the maximum magnitude of its nontrivial aperiodic correlation, denoted as $\theta_{\max} ({\mathcal S})$. The lower bound on $\theta_{\max} ({\mathcal S})$ was originally presented by Welch, and improved later by Levenshtein. In this paper, a Fourier transform approach is introduced in an attempt to improve the Levenshtein's lower bound. Through the approach, a new expression of the Levenshtein bound is developed. Along with numerical supports, it is found that $\theta_{\max} ^2 ({\mathcal S}) > 0.3584 n-0.0810$ for $M=3$ and $n \ge 4$, and $\theta_{\max} ^2 ({\mathcal S}) > 0.4401 n-0.1053$ for $M=4$ and $n \ge 4$, respectively, which are tighter than the original Welch and Levenshtein bounds.

Based on the cyclotomy classes of extension fields, a family of binary cyclotomic sequences are constructed and their pseudorandom measures (i.e., the well-distribution measure and the correlation measure of order k) are estimated using certain exponential sums. A lower bound on the linear complexity profile is also presented in terms of the correlation measure.

In this correspondence, we present a family of binary 2n sequences of period 2(2n-1) where n is an integer, which can be seen as a generalization of nonlinear binary sequences obtained from Z4 sequences and recently constructed GKW (Gold, Kasami, and Welch)-like sequences. The sequences have low correlations and are useful in code-division multiple-access (CDMA) communication systems and cryptography

Families of binary sequences are recursively defined, with the first family consisting of a single reference m-sequence. The crosscorrelation function of any of these sequences with the reference m-sequence has a unique pattern which determines a code word corresponding to the given sequence. One suggested application of the technique is to a data transmission system using modulation of an m-sequence carrier.

The well known family of binary twin-prime sequences is generalised to the multiple-valued case by employing a polyphase representation of the sequence elements. These polyphase versions exhibit similar periodic and aperiodic auto-correlation properties to their binary counterparts, and are referred to as q-phase related-prime (RP) sequences. These sequences have length L=r·s, for r and s both prime, and with s=r+k. They are constructed by combining two polyphase Legendre sequences of lengths r and s, and modifying the resulting composite sequence at certain points. A two-dimensional array structure is employed in the construction and analysis of these sequences. The original q-phase Legendre sequences are derived by converting the index sequences of lengths r and s to modulo-q form. When q is even, two classes of RP sequence arise, depending on whether L≡q+1 mod 2q or L≡1 mod 2q. For odd q, only a single class is available, and here L≡1 mod 2q. The out-of-phase periodic correlation values of these RP sequences are independent of the sequence length, and depend only on the number of phases q and the difference k between the two related primes. The maximum out-of-phase correlation values is given by 1−k. Tables of available sequences are presented.