397 research outputs found
Capacity-Achieving Ensembles of Accumulate-Repeat-Accumulate Codes for the Erasure Channel with Bounded Complexity
The paper introduces ensembles of accumulate-repeat-accumulate (ARA) codes
which asymptotically achieve capacity on the binary erasure channel (BEC) with
{\em bounded complexity}, per information bit, of encoding and decoding. It
also introduces symmetry properties which play a central role in the
construction of capacity-achieving ensembles for the BEC with bounded
complexity. The results here improve on the tradeoff between performance and
complexity provided by previous constructions of capacity-achieving ensembles
of codes defined on graphs. The superiority of ARA codes with moderate to large
block length is exemplified by computer simulations which compare their
performance with those of previously reported capacity-achieving ensembles of
LDPC and IRA codes. The ARA codes also have the advantage of being systematic.Comment: Submitted to IEEE Trans. on Information Theory, December 1st, 2005.
Includes 50 pages and 13 figure
Capacity-Achieving Codes with Bounded Graphical Complexity on Noisy Channels
We introduce a new family of concatenated codes with an outer low-density
parity-check (LDPC) code and an inner low-density generator matrix (LDGM) code,
and prove that these codes can achieve capacity under any memoryless
binary-input output-symmetric (MBIOS) channel using maximum-likelihood (ML)
decoding with bounded graphical complexity, i.e., the number of edges per
information bit in their graphical representation is bounded. In particular, we
also show that these codes can achieve capacity on the binary erasure channel
(BEC) under belief propagation (BP) decoding with bounded decoding complexity
per information bit per iteration for all erasure probabilities in (0, 1). By
deriving and analyzing the average weight distribution (AWD) and the
corresponding asymptotic growth rate of these codes with a rate-1 inner LDGM
code, we also show that these codes achieve the Gilbert-Varshamov bound with
asymptotically high probability. This result can be attributed to the presence
of the inner rate-1 LDGM code, which is demonstrated to help eliminate high
weight codewords in the LDPC code while maintaining a vanishingly small amount
of low weight codewords.Comment: 17 pages, 2 figures. This paper is to be presented in the 43rd Annual
Allerton Conference on Communication, Control and Computing, Monticello, IL,
USA, Sept. 28-30, 200
Capacity-achieving ensembles for the binary erasure channel with bounded complexity
We present two sequences of ensembles of non-systematic irregular
repeat-accumulate codes which asymptotically (as their block length tends to
infinity) achieve capacity on the binary erasure channel (BEC) with bounded
complexity per information bit. This is in contrast to all previous
constructions of capacity-achieving sequences of ensembles whose complexity
grows at least like the log of the inverse of the gap (in rate) to capacity.
The new bounded complexity result is achieved by puncturing bits, and allowing
in this way a sufficient number of state nodes in the Tanner graph representing
the codes. We also derive an information-theoretic lower bound on the decoding
complexity of randomly punctured codes on graphs. The bound holds for every
memoryless binary-input output-symmetric channel and is refined for the BEC.Comment: 47 pages, 9 figures. Submitted to IEEE Transactions on Information
Theor
Spatially-Coupled MacKay-Neal Codes and Hsu-Anastasopoulos Codes
Kudekar et al. recently proved that for transmission over the binary erasure
channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the
coupled ensemble to the MAP threshold of the underlying LDPC codes. One major
drawback of the capacity-achieving spatially-coupled LDPC codes is that one
needs to increase the column and row weight of parity-check matrices of the
underlying LDPC codes.
It is proved, that Hsu-Anastasopoulos (HA) codes and MacKay-Neal (MN) codes
achieve the capacity of memoryless binary-input symmetric-output channels under
MAP decoding with bounded column and row weight of the parity-check matrices.
The HA codes and the MN codes are dual codes each other.
The aim of this paper is to present an empirical evidence that
spatially-coupled MN (resp. HA) codes with bounded column and row weight
achieve the capacity of the BEC. To this end, we introduce a spatial coupling
scheme of MN (resp. HA) codes. By density evolution analysis, we will show that
the resulting spatially-coupled MN (resp. HA) codes have the BP threshold close
to the Shannon limit.Comment: Corrected typos in degree distributions \nu and \mu of MN and HA
code
Fingerprinting with Minimum Distance Decoding
This work adopts an information theoretic framework for the design of
collusion-resistant coding/decoding schemes for digital fingerprinting. More
specifically, the minimum distance decision rule is used to identify 1 out of t
pirates. Achievable rates, under this detection rule, are characterized in two
distinct scenarios. First, we consider the averaging attack where a random
coding argument is used to show that the rate 1/2 is achievable with t=2
pirates. Our study is then extended to the general case of arbitrary
highlighting the underlying complexity-performance tradeoff. Overall, these
results establish the significant performance gains offered by minimum distance
decoding as compared to other approaches based on orthogonal codes and
correlation detectors. In the second scenario, we characterize the achievable
rates, with minimum distance decoding, under any collusion attack that
satisfies the marking assumption. For t=2 pirates, we show that the rate
is achievable using an ensemble of random linear
codes. For , the existence of a non-resolvable collusion attack, with
minimum distance decoding, for any non-zero rate is established. Inspired by
our theoretical analysis, we then construct coding/decoding schemes for
fingerprinting based on the celebrated Belief-Propagation framework. Using an
explicit repeat-accumulate code, we obtain a vanishingly small probability of
misidentification at rate 1/3 under averaging attack with t=2. For collusion
attacks which satisfy the marking assumption, we use a more sophisticated
accumulate repeat accumulate code to obtain a vanishingly small
misidentification probability at rate 1/9 with t=2. These results represent a
marked improvement over the best available designs in the literature.Comment: 26 pages, 6 figures, submitted to IEEE Transactions on Information
Forensics and Securit
Multiplicatively Repeated Non-Binary LDPC Codes
We propose non-binary LDPC codes concatenated with multiplicative repetition
codes. By multiplicatively repeating the (2,3)-regular non-binary LDPC mother
code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9,
1/12,... Surprisingly, such simple low-rate non-binary LDPC codes outperform
the best low-rate binary LDPC codes so far. Moreover, we propose the decoding
algorithm for the proposed codes, which can be decoded with almost the same
computational complexity as that of the mother code.Comment: To appear in IEEE Transactions on Information Theor
On Universal Properties of Capacity-Approaching LDPC Ensembles
This paper is focused on the derivation of some universal properties of
capacity-approaching low-density parity-check (LDPC) code ensembles whose
transmission takes place over memoryless binary-input output-symmetric (MBIOS)
channels. Properties of the degree distributions, graphical complexity and the
number of fundamental cycles in the bipartite graphs are considered via the
derivation of information-theoretic bounds. These bounds are expressed in terms
of the target block/ bit error probability and the gap (in rate) to capacity.
Most of the bounds are general for any decoding algorithm, and some others are
proved under belief propagation (BP) decoding. Proving these bounds under a
certain decoding algorithm, validates them automatically also under any
sub-optimal decoding algorithm. A proper modification of these bounds makes
them universal for the set of all MBIOS channels which exhibit a given
capacity. Bounds on the degree distributions and graphical complexity apply to
finite-length LDPC codes and to the asymptotic case of an infinite block
length. The bounds are compared with capacity-approaching LDPC code ensembles
under BP decoding, and they are shown to be informative and are easy to
calculate. Finally, some interesting open problems are considered.Comment: Published in the IEEE Trans. on Information Theory, vol. 55, no. 7,
pp. 2956 - 2990, July 200
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