3,647 research outputs found
Shift-Inequivalent Decimations of the Sidelnikov-Lempel-Cohn-Eastman Sequences
We consider the problem of finding maximal sets of shift-inequivalent
decimations of Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences (as well as the
equivalent problem of determining the multiplier groups of the almost
difference sets associated with these sequences). We derive a numerical
necessary condition for a residue to be a multiplier of an SLCE almost
difference set. Using our necessary condition, we show that if is an odd
prime and is an SLCE almost difference set over then the
multiplier group of is trivial. Consequently, for each odd prime we
obtain a family of shift-inequivalent balanced periodic sequences
(where is the Euler-Totient function) each having period and
nearly perfect autocorrelation
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
Autocorrelation-Run Formula for Binary Sequences
The autocorrelation function and the run structure are two basic notions for
binary sequences, and have been used as two independent postulates to test
randomness of binary sequences ever since Golomb 1955. In this paper, we prove
for binary sequence that the autocorrelation function is in fact completely
determined by its run structure.Comment: 18 pages, 1 figur
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
The Proof of Lin's Conjecture via the Decimation-Hadamard Transform
In 1998, Lin presented a conjecture on a class of ternary sequences with
ideal 2-level autocorrelation in his Ph.D thesis. Those sequences have a very
simple structure, i.e., their trace representation has two trace monomial
terms. In this paper, we present a proof for the conjecture. The mathematical
tools employed are the second-order multiplexing decimation-Hadamard transform,
Stickelberger's theorem, the Teichm\"{u}ller character, and combinatorial
techniques for enumerating the Hamming weights of ternary numbers. As a
by-product, we also prove that the Lin conjectured ternary sequences are
Hadamard equivalent to ternary -sequences
Large Families of Optimal Two-Dimensional Optical Orthogonal Codes
Nine new 2-D OOCs are presented here, all sharing the common feature of a
code size that is much larger in relation to the number of time slots than
those of constructions appearing previously in the literature. Each of these
constructions is either optimal or asymptotically optimal with respect to
either the original Johnson bound or else a non-binary version of the Johnson
bound introduced in this paper.
The first 5 codes are constructed using polynomials over finite fields - the
first construction is optimal while the remaining 4 are asymptotically optimal.
The next two codes are constructed using rational functions in place of
polynomials and these are asymptotically optimal. The last two codes, also
asymptotically optimal, are constructed by composing two of the above codes
with a constant weight binary code.
Also presented, is a three-dimensional OOC that exploits the polarization
dimension.
Finally, phase-encoded optical CDMA is considered and construction of two
efficient codes are provided
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
Impact of Different Spreading Codes Using FEC on DWT Based MC-CDMA System
The effect of different spreading codes in DWT based MC-CDMA wireless
communication system is investigated. In this paper, we present the Bit Error
Rate (BER) performance of different spreading codes (Walsh-Hadamard code,
Orthogonal gold code and Golay complementary sequences) using Forward Error
Correction (FEC) of the proposed system. The data is analyzed and is compared
among different spreading codes in both coded and uncoded cases. It is found
via computer simulation that the performance of the proposed coded system is
much better than that of the uncoded system irrespective of the spreading codes
and all the spreading codes show approximately similar nature for both coded
and uncoded in all modulation schemes.Comment: 10 Pages; International Journal of Mobile Network Communications &
Telematics (IJMNCT) Vol.2, No.3, June 201
On Binary Cyclic Codes with Five Nonzero Weights
Let , , be odd and .
In this paper the value distribution of following exponential sums
\sum\limits_{x\in \bF_q}(-1)^{\mathrm{Tr}_1^n(\alpha x^{2^{2k}+1}+\beta
x^{2^k+1}+\ga x)}\quad(\alpha,\beta,\ga\in \bF_{q}) is determined. As an
application, the weight distribution of the binary cyclic code \cC, with
parity-check polynomial where , and
are the minimal polynomials of , and
respectively for a primitive element of \bF_q, is
also determined
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