On the linear complexity of binary lattices

Acta Arithmetica, 2010

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.

Discrete Mathematics, 2012

This paper concerns the study of the correlation measures of finite binary sequences, more particularily the dependence of correlation measures of even order and correlation measures of odd order. These results generalize previous results due to Gyarmati [7] and to Anantharam [3] and provide a partial answer to a conjecture due to Mauduit [12]. The last part of the paper concerns the generalization of this study to the case of finite binary n-dimensional lattices.

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2023

In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity, correlation measure of order , expansion complexity and 2-adic complexity. The number-theoretic sequences are the Legendre sequence and the two-prime generator, the Thue-Morse sequence and its sub-sequence along squares, and the prime omega sequences for integers and polynomials.

The Ramanujan Journal, 2014

We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al. in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family

Lecture Notes in Computer Science, 2018

The cross-combined measure (which is a natural extension of cross-correlation measure) is introduced and important constructions of large families of binary lattices with optimal or nearly optimal cross-combined measures are presented. These results are also strongly related to the onedimensional case: An easy method is showed obtaining strong constructions of families of binary sequences with nearly optimal cross-correlation measures based on the previous constructions of families of lattices. The important feature of this result is that so far there exists only one type of constructions of very large families of binary sequences with small cross-correlation measure, and this only type of constructions was based on one-variable irreducible polynomials. Since it is very complicated to construct one-variable irreducible polynomials over Fp, it became necessary to show other types of constructions where the generation of sequences is much faster. Using binary lattices based on two-variable irreducible polynomials this problem can be avoided. (Since, contrary to one-variable polynomials, using Schöneman-Eisenstein criteria it is possible to generate two-variable irreducible polynomials over Fp fast.)

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.

Applied Algebra and Number Theory

Large families of binary sequences of the same length are considered and a new measure, the cross-correlation measure of order k is introduced to study the connection between the sequences belonging to the family. It is shown that this new measure is related to certain other important properties of families of binary sequences. Then the size of the cross-correlation measure is studied. Finally, the cross-correlation measures of two important families of pseudorandom binary sequences are estimated.

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.