New Family of Binary Sequences with Low Correlation

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).

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.

The Gold sequence family [1] is one of the oldest and most well‐known families of binary sequences with optimal correlation properties. This chapter focuses on the binary Gold sequences and their relatives called Gold‐like sequences. Mathematical representations as well as the family sizes and maximum correlation magnitudes of these sequences are reviewed.

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.

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

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.

For odd prime p, two new p-ary sequence families with period pn−1 are constructed for odd n=2l+1=me and even n=2l=me respectively. It is shown that, for a given integer ρ, the proposed sequence families have maximum correlation 1+pn/2+(ρ−1)e+1, family size (pn−1)pnρ/2, and maximum linear span n(n+3).

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.

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.

In recent papers [14], [15] I studied collision and avalanche effect in families of finite pseudorandom binary sequences. Motivated by applications, Mauduit and Sarkozy in [13] generalized and extended this theory from the binary case to k-ary sequences, i.e., to k symbols. They constructed a large family of k-ary sequences with strong pseudorandom properties. In this paper our goal is to extend the study of the pseudorandom properties mentioned above to k-ary sequences. The aim of this paper is twofold. First we will extend the definitions of collision and avalanche effect to k-ary sequences, and then we will study these related properties in a large family of pseudorandom k-ary sequences with “small” pseudorandom measures.

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.

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.

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.

Self-similarity of a family of binary quasiperiodic sequences associated with quadratic irrationals

This paper presents the first concerted look at low correlation sequence families over quadrature amplitude modulation (QAM) constellations of size and their potential applicability as spreading sequences in a code-division multiple-access (CDMA) setting. Five constructions are presented, and it is shown how such sequence families have the ability to transport a larger amount of data as well as enable variable-rate signaling on the reverse link. Canonical family has period , normalized maximum-correlation parameter bounded above by , where ranges from in the 16-QAM case to for large . In a CDMA setting, each user is enabled to transfer bits of data per period of the spreading sequence which can be increased to bits of data by halving the size of the sequence family. The technique used to construct is easily extended to produce larger sequence families and an example is provided. Selected family has a lower value of but permits only -bit data modulation. The interleaved 16-QAM sequence family, has and supports 3-bit data modulation. The remaining two families are over a quadrature-pulse amplitude modulation (Q-PAM) subset of size of the -QAM constellation. Family has a lower value of in comparison with Family , while still permitting -bit data modulation. Interleaved Family , over the 8-ary Q-PAM constellation, permits 3-bit data modulation and interestingly achieves the Welch lower bound on .