The cross-correlation measure for families of binary sequences

arXiv (Cornell University), 2021

Correlation measure of order k is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of order k and another two pseudorandom measures: the N th linear complexity and the N th maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it.

IEEE Transactions on Information Theory, 2000

Considered is the distribution of the cross correlation between m-sequences of length 2 m 0 1, where m = 2k, and msequences of shorter length 2 k 01. New pairs of m-sequences with three-valued cross correlation are found and the complete correlation distribution is determined. Finally, we conjecture that there are no more cases with a three-valued cross correlation apart from the ones proven here.

2002

In this paper, we consider the problem of evaluating the complexity of random and pseudo random binary sequences. It has been shown that the traditional estimation of linear complexity does not allow us to fully estimate of the probability of forward and backward prediction of a sequence. In this paper, we introduce the concept of maximum complexity order. In particular, the estimation of the mathematical expectation of the maximum order complexity for quite random sequences has been obtained. A fast algorithm is suggested for determining the maximum order complexity of sequences with length n. The time that is required is proportional to n⋅log2n much less when compared to Berlekamp-Massey’s algorithm, whose time consumption is in proportion to n. It has been shown that the characteristics of the feedback Boolean function make it possible to estimate the direct and reverse predictability of the sequence being tested.

Combinatorics, Probability & Computing, 2006

Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences EN ∈ {−1, 1} N in order to measure their 'level of randomness'. Two of these parameters are the normality measure N (EN ) and the correlation measure C k (EN ) of order k, which focus on different combinatorial aspects of EN . In their work, amongst others, Mauduit and Sárközy investigated the minimal possible value of these parameters.

Journal of applied mathematics & informatics, 2013

The design of large family size with the optimal cross-correlation property is important in spread spectrum and code division multiple access communication systems. In this paper we present the sequences with the decimation d = 2 • 2 m − 1, calculate the cross-correlation spectrum for 0 ≤ t ≤ 2 n − 2 and count the number of the value 2 m − 1 occurring for 0 ≤ τ ≤ 2 n − 2. The sequences have the optimal cross-correlation property. The work on this paper can make it easier to count the number of the whole value occurring for 0 ≤ τ ≤ 2 n − 2.

IEEE Transactions on Information Theory

The (classical) crosscorrelation is an important measure of pseudorandomness of two binary sequences for applications in communications. The arithmetic crosscorrelation is another figure of merit introduced by Goresky and Klapper generalizing Mandelbaum's arithmetic autocorrelation. First we observe that the arithmetic crosscorrelation is constant for two binary sequences of coprime periods which complements the analogous result for the classical crosscorrelation. Then we prove upper bounds for the constant arithmetic crosscorrelation of two Legendre sequences of different periods and of two binary m-sequences of coprime periods, respectively.

IEEE International Symposium on Spread Spectrum Techniques and Applications, 1996

M-sequences have been studied extensively as the nearest approximation to random sequences. In general, the problem of computing crosscorrelations between m-sequences by analytical techniques has been declared untractable. In this research, cross-correlations for all sequences of lengths up to 218-1 have been examined numerically, in the hope of finding some predictable patterns. One pattern which emerged from this numerical analysis was the existence of cross-correlation peaks well in excess of those predicted by statistical techniques [l]. This paper demonstrates these "anomalous" peaks to be due finite algebra effects. These results suggest the existence of some algebraically computable correlations, apart from those already known from Galois Field theory [2], [SI. The technique developed here can be used to determine the pairs of sequences with high crosscorrelation peaks, the approximate value of these peaks and the relative phasing of the sequences. Elimination of such pairs of sequences results in a dramatic reduction in the peak cross-correlation among the set of remaining sequences. These have been named "Constrained Connected Sets". When used in conjunction with sequences whose crosscorrelations are predictable by Galois Field analysis [2], [SI, this technique may prove useful in the design of code families to meet specific crosscorrelation requirements.

2004

New binary and ternary sequences with low correlation and simple implementation are presented. The sequences are unfolded from arrays, whose columns are cyclic shifts of a short sequence or constant columns and whose shift sequence (sequence of column shifts) has the distinct difference property. It is known that a binary m-sequence/GMW sequence of length 2 $^{\rm 2{\it m}}$ – 1 can be folded row-by-row into an array of 2m – 1 rows of length 2m + 1. We use this to construct new arrays which have at most one column matching for any two dimensional cyclic shift and therefore have low off-peak autocorrelation. The columns of the array can be multiplied by binary orthogonal sequences of commensurate length to produce a set of arrays with low cross-correlation. These arrays are unfolded to produce sequence sets with identical low correlation.

Discrete Mathematics, 2009

We extend the results of Goubin, Mauduit and Sárközy on the well-distribution measure and the correlation measure of order k of the sequence of Legendre sequences with polynomial argument in several ways. We analyze sequences of quadratic characters of finite fields of prime power order and consider in each case two, in general, different definitions of well-distribution measure and correlation measure of order k, respectively.

2017

We prove a relation between two measures of pseudorandomness, the arithmetic autocorrelation, and the correlation measure of order k. Roughly speaking, we show that any binary sequence with small correlation measure of order k up to a sufficiently large k cannot have a large arithmetic correlation. We apply our result to several classes of sequences including Legendre sequences defined with polynomials.