On the correlation of binary sequences, II

Acta Arithmetica, 2008

Israel Journal of Mathematics, 2007

We consider a system of "generalised linear forms" defined on a subset x = (x ij) of R d by L 1 (x)(k) = d 1 j=1 g k 1j (x 1j),. .. , L l (x)(k) = d l j=1 g k lj (x lj) ∈ R, for k ≥ 1,

Finite Fields and Their Applications, 2001

Binary m-sequences are widely applied in navigation, radar, and communication systems because of their nice autocorrelation and cross-correlation properties. In this paper, we consider the cross-correlation between a binary m-sequence of length 2K!1 and a decimation of that sequence by an integer t. We will be interested in the number of values attained by such cross-correlations. As is well known, this number equals the number of nonzero weights in the dual of the binary cyclic code C R of length 2K!1 with de"ning zeros and R, where is a primitive element in GF(2K). There are many pairs (m, t) for which C, R is known or conjectured to have only few nonzero weights. The three-weight examples include the following cases: (a) t"1#2P, if m/(m, r) odd, (b) t"2P!2P#1, if m/(m, r) odd, (c) m"2r#1 odd, t"2P#3, and (d) m odd, 4r,!1 mod m, t"2P#2P!1. We present a method of proving many of these known or conjectured results, including all of the above cases, in a uni"ed way.

2011

Binary perfect sequences and their variations have applications in various areas such as signal processing, synchronizing and distance measuring radars. This survey discusses their p-ary analogs, other variations and related matters. Many new results are also presented. Introduction: In recent years there have been many publications on time-discrete one and twodimensional sequences and arrays with perfect autocorrelation functions. Such sequences find applications in signal processing and as aperture functions for electromagnetic and acoustic imaging. Applications of two-dimensional perfect binary arrays are found in 2-D synchronization (Hershey & Yarlagadda (1983)) and timefrequency coding (Golomb & Taylor (1982)). In his invited address at the 1991 British Combinatorial Conference, Golomb gave an excellent exposition on why “small correlations” of sequences and arrays are desirable in dealing with radar problems (Golomb (1991)). Some fundamental results on sequences with small cor...

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.

The Ramanujan Journal, 2013

The linear complexity is an important and frequently used measure of unpredictability and pseudorandomness of binary sequences. In this paper our goal is to extend this notion to two dimensions. We will define and study the linear complexity of binary lattices. The linear complexity of a truly random binary lattice will be estimated. Finally, we will analyze the connection between the linear complexity and the correlation measures, and we will utilize the inequalities obtained in this way for estimating the linear complexity of an important special binary lattice. Finally, we will study the connection between the linear complexity of binary lattices and of the associated binary sequences.

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

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.

Publicationes Mathematicae Debrecen, 2011

In the last 15 years a new constructive theory of pseudorandomness of binary sequences has been developed. Later this theory was extended to n dimensions, i.e., to the study of pseudorandomness of binary lattices. In the applications it is not enough to consider single binary sequences, one also needs information on the structure of large families of binary sequences with strong pseudorandom properties. Thus the related notions of family complexity, collision and avalanche effect have been introduced. In this paper our goal is to extend these definitions to binary lattices, and we will present constructions of large families of binary lattices with strong pseudorandom properties such that these families also possess a nice structure.