3,091 research outputs found
Design and Analysis of Nonbinary LDPC Codes for Arbitrary Discrete-Memoryless Channels
We present an analysis, under iterative decoding, of coset LDPC codes over
GF(q), designed for use over arbitrary discrete-memoryless channels
(particularly nonbinary and asymmetric channels). We use a random-coset
analysis to produce an effect that is similar to output-symmetry with binary
channels. We show that the random selection of the nonzero elements of the
GF(q) parity-check matrix induces a permutation-invariance property on the
densities of the decoder messages, which simplifies their analysis and
approximation. We generalize several properties, including symmetry and
stability from the analysis of binary LDPC codes. We show that under a Gaussian
approximation, the entire q-1 dimensional distribution of the vector messages
is described by a single scalar parameter (like the distributions of binary
LDPC messages). We apply this property to develop EXIT charts for our codes. We
use appropriately designed signal constellations to obtain substantial shaping
gains. Simulation results indicate that our codes outperform multilevel codes
at short block lengths. We also present simulation results for the AWGN
channel, including results within 0.56 dB of the unconstrained Shannon limit
(i.e. not restricted to any signal constellation) at a spectral efficiency of 6
bits/s/Hz.Comment: To appear, IEEE Transactions on Information Theory, (submitted
October 2004, revised and accepted for publication, November 2005). The
material in this paper was presented in part at the 41st Allerton Conference
on Communications, Control and Computing, October 2003 and at the 2005 IEEE
International Symposium on Information Theor
Deriving Good LDPC Convolutional Codes from LDPC Block Codes
Low-density parity-check (LDPC) convolutional codes are capable of achieving
excellent performance with low encoding and decoding complexity. In this paper
we discuss several graph-cover-based methods for deriving families of
time-invariant and time-varying LDPC convolutional codes from LDPC block codes
and show how earlier proposed LDPC convolutional code constructions can be
presented within this framework. Some of the constructed convolutional codes
significantly outperform the underlying LDPC block codes. We investigate some
possible reasons for this "convolutional gain," and we also discuss the ---
mostly moderate --- decoder cost increase that is incurred by going from LDPC
block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010;
revised August 2010, revised November 2010 (essentially final version).
(Besides many small changes, the first and second revised versions contain
corrected entries in Tables I and II.
Convolutional and tail-biting quantum error-correcting codes
Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to
4096 states and minimum distances up to 10 are constructed as stabilizer codes
from classical self-orthogonal rate-1/n F_4-linear and binary linear
convolutional codes, respectively. These codes generally have higher rate and
less decoding complexity than comparable quantum block codes or previous
quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same
rate and error-correction capability and essentially the same decoding
algorithms are derived from these convolutional codes via tail-biting.Comment: 30 pages. Submitted to IEEE Transactions on Information Theory. Minor
revisions after first round of review
Lower bounds for constant query affine-invariant LCCs and LTCs
Affine-invariant codes are codes whose coordinates form a vector space over a
finite field and which are invariant under affine transformations of the
coordinate space. They form a natural, well-studied class of codes; they
include popular codes such as Reed-Muller and Reed-Solomon. A particularly
appealing feature of affine-invariant codes is that they seem well-suited to
admit local correctors and testers.
In this work, we give lower bounds on the length of locally correctable and
locally testable affine-invariant codes with constant query complexity. We show
that if a code is an -query
locally correctable code (LCC), where is a finite field and
is a finite alphabet, then the number of codewords in is
at most . Also, we show that if
is an -query locally testable
code (LTC), then the number of codewords in is at most
. The dependence on in these
bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan
(ITCS `13) construct affine-invariant codes via lifting that have the same
asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas
previously, Ben-Sasson and Sudan (RANDOM `11) assumed linearity to derive
similar results.
Our analysis uses higher-order Fourier analysis. In particular, we show that
the codewords corresponding to an affine-invariant LCC/LTC must be far from
each other with respect to Gowers norm of an appropriate order. This then
allows us to bound the number of codewords, using known decomposition theorems
which approximate any bounded function in terms of a finite number of
low-degree non-classical polynomials, upto a small error in the Gowers norm
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