26,212 research outputs found
Sequences with Low Correlation
Pseudorandom sequences are used extensively in communications and remote
sensing. Correlation provides one measure of pseudorandomness, and low
correlation is an important factor determining the performance of digital
sequences in applications. We consider the problem of constructing pairs
of sequences such that both and have low mean square
autocorrelation and and have low mean square mutual crosscorrelation.
We focus on aperiodic correlation of binary sequences, and review recent
contributions along with some historical context.Comment: 24 page
Advances in the merit factor problem for binary sequences
The identification of binary sequences with large merit factor (small
mean-squared aperiodic autocorrelation) is an old problem of complex analysis
and combinatorial optimization, with practical importance in digital
communications engineering and condensed matter physics. We establish the
asymptotic merit factor of several families of binary sequences and thereby
prove various conjectures, explain numerical evidence presented by other
authors, and bring together within a single framework results previously
appearing in scattered form. We exhibit, for the first time, families of
skew-symmetric sequences whose asymptotic merit factor is as large as the best
known value (an algebraic number greater than 6.34) for all binary sequences;
this is interesting in light of Golay's conjecture that the subclass of
skew-symmetric sequences has asymptotically optimal merit factor. Our methods
combine Fourier analysis, estimation of character sums, and estimation of the
number of lattice points in polyhedra.Comment: 31 pages, minor revision
Note on the Merit Factors of Sequences
A method for estimating the merit factors of sequences will be provided. The
result is also effective in determining the nonexistence of certain infinite
collections of cyclic difference sets and cyclic matrices and associated binary
sequences.Comment: Key Words: Binary sequences, Merit factor, Barker sequence, cyclic
Hadamard matrix, Menon difference sets. 19 page
Modifications on Character Sequences and Construction of Large Even Length Binary Sequences
It has been noticed that all the known binary sequences having the asymptotic
merit factor are the modifications to the real primitive characters. In
this paper, we give a new modification of the character sequences at length
, where 's are distinct odd primes and is finite.
Based on these new modifications, for with 's distinct
odd primes, we can construct a binary sequence of length with asymptotic
merit factor $6.0
Low Correlation Sequences from Linear Combinations of Characters
Pairs of binary sequences formed using linear combinations of multiplicative
characters of finite fields are exhibited that, when compared to random
sequence pairs, simultaneously achieve significantly lower mean square
autocorrelation values (for each sequence in the pair) and significantly lower
mean square crosscorrelation values. If we define crosscorrelation merit factor
analogously to the usual merit factor for autocorrelation, and if we define
demerit factor as the reciprocal of merit factor, then randomly selected binary
sequence pairs are known to have an average crosscorrelation demerit factor of
. Our constructions provide sequence pairs with crosscorrelation demerit
factor significantly less than , and at the same time, the autocorrelation
demerit factors of the individual sequences can also be made significantly less
than (which also indicates better than average performance). The sequence
pairs studied here provide combinations of autocorrelation and crosscorrelation
performance that are not achievable using sequences formed from single
characters, such as maximal linear recursive sequences (m-sequences) and
Legendre sequences. In this study, exact asymptotic formulae are proved for the
autocorrelation and crosscorrelation merit factors of sequence pairs formed
using linear combinations of multiplicative characters. Data is presented that
shows that the asymptotic behavior is closely approximated by sequences of
modest length.Comment: 47 page
Aperiodic Crosscorrelation of Sequences Derived from Characters
It is shown that pairs of maximal linear recursive sequences (m-sequences)
typically have mean square aperiodic crosscorrelation on par with that of
random sequences, but that if one takes a pair of m-sequences where one is the
reverse of the other, and shifts them appropriately, one can get significantly
lower mean square aperiodic crosscorrelation. Sequence pairs with even lower
mean square aperiodic crosscorrelation are constructed by taking a Legendre
sequence, cyclically shifting it, and then cutting it (approximately) in half
and using the halves as the sequences of the pair. In some of these
constructions, the mean square aperiodic crosscorrelation can be lowered
further if one truncates or periodically extends (appends) the sequences. Exact
asymptotic formulae for mean squared aperiodic crosscorrelation are proved for
sequences derived from additive characters (including m-sequences and modified
versions thereof) and multiplicative characters (including Legendre sequences
and their relatives). Data is presented that shows that sequences of modest
length have performance that closely approximates the asymptotic formulae.Comment: 54 page
Rudin-Shapiro-Like Sequences with Maximum Asymptotic Merit Factor
Borwein and Mossinghoff investigated the Rudin-Shapiro-like sequences, which
are infinite families of binary sequences, usually represented as polynomials.
Each family of Rudin-Shapiro-like sequences is obtained from a starting
sequence (which we call the seed) by a recursive construction that doubles the
length of the sequence at each step, and many sequences produced in this manner
have exceptionally low aperiodic autocorrelation. Borwein and Mossinghoff
showed that the asymptotic autocorrelation merit factor for any such family is
at most , and found the seeds of length or less that produce the
maximum asymptotic merit factor of . The definition of Rudin-Shapiro-like
sequences was generalized by Katz, Lee, and Trunov to include sequences with
arbitrary complex coefficients, among which are families of low autocorrelation
polyphase sequences. Katz, Lee, and Trunov proved that the maximum asymptotic
merit factor is also for this larger class. Here we show that a family of
such Rudin-Shapiro-like sequences achieves asymptotic merit factor if and
only if the seed is either of length or is the interleaving of a pair of
Golay complementary sequences. For small seed lengths where this is not
possible, the optimal seeds are interleavings of pairs that are as close as
possible to being complementary pairs, and the idea of an almost-complementary
pair makes sense of remarkable patterns in previously unexplained data on
optimal seeds for binary Rudin-Shapiro-like sequences.Comment: 22 page
Crosscorrelation of Rudin-Shapiro-Like Polynomials
We consider the class of Rudin-Shapiro-like polynomials, whose norms on
the complex unit circle were studied by Borwein and Mossinghoff. The polynomial
is identified with the sequence
of its coefficients. From the norm of a
polynomial, one can easily calculate the autocorrelation merit factor of its
associated sequence, and conversely. In this paper, we study the
crosscorrelation properties of pairs of sequences associated to
Rudin-Shapiro-like polynomials. We find an explicit formula for the
crosscorrelation merit factor. A computer search is then used to find pairs of
Rudin-Shapiro-like polynomials whose autocorrelation and crosscorrelation merit
factors are simultaneously high. Pursley and Sarwate proved a bound that limits
how good this combined autocorrelation and crosscorrelation performance can be.
We find infinite families of polynomials whose performance approaches quite
close to this fundamental limit.Comment: 32 page
Family complexity and cross-correlation measure for families of binary sequences
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
, , of binary sequences
of length in terms of the cross-correlation measure of its dual family
, .
We apply this result to the family of sequences of Legendre symbols with
irreducible quadratic polynomials modulo with middle coefficient , that
is, for
, where is a quadratic nonresidue modulo , showing
that this family as well as its dual family have both a large family complexity
and a small cross-correlation measure up to a rather large order
Merit factors of polynomials derived from difference sets
The problem of constructing polynomials with all coefficients or and
large merit factor (equivalently with small norm on the unit circle)
arises naturally in complex analysis, condensed matter physics, and digital
communications engineering. Most known constructions arise (sometimes in a
subtle way) from difference sets, in particular from Paley and Singer
difference sets. We consider the asymptotic merit factor of polynomials
constructed from other difference sets, providing the first essentially new
examples since 1991. In particular we prove a general theorem on the asymptotic
merit factor of polynomials arising from cyclotomy, which includes results on
Hall and Paley difference sets as special cases. In addition, we establish the
asymptotic merit factor of polynomials derived from Gordon-Mills-Welch
difference sets and Sidelnikov almost difference sets, proving two recent
conjectures.Comment: 22 pages, this revision contains a more general version of Thm. 2.
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