3,038 research outputs found
On the Complexity of Exact Maximum-Likelihood Decoding for Asymptotically Good Low Density Parity Check Codes: A New Perspective
The problem of exact maximum-likelihood (ML) decoding of general linear codes is well-known to be NP-hard. In this paper, we show that exact ML decoding of a class of asymptotically good low density parity check codes — expander codes — over binary symmetric channels (BSCs) is possible with an average-case polynomial complexity. This offers a new way of looking at the complexity issue of exact ML decoding for communication systems where the randomness in channel plays a fundamental central role. More precisely, for any bit-flipping probability p in a nontrivial range, there exists a rate region of non-zero support and a family of asymptotically good codes which achieve error probability exponentially decaying in coding length n while admitting exact ML decoding in average-case polynomial time. As p approaches zero, this rate region approaches the Shannon channel capacity region. Similar results can be extended to AWGN channels, suggesting it may be feasible to eliminate the error floor phenomenon associated with belief-propagation decoding of LDPC codes in the high SNR regime. The derivations are based on a hierarchy of ML certificate decoding algorithms adaptive to the channel realization. In this process, we propose an efficient O(n^2) new ML certificate algorithm based on the max-flow algorithm. Moreover, exact ML decoding of the considered class of codes constructed from LDPC codes with regular left degree, of which the considered expander codes are a special case, remains NP-hard; thus giving an interesting contrast between the worst-case and average-case complexities
Iterative Algebraic Soft-Decision List Decoding of Reed-Solomon Codes
In this paper, we present an iterative soft-decision decoding algorithm for
Reed-Solomon codes offering both complexity and performance advantages over
previously known decoding algorithms. Our algorithm is a list decoding
algorithm which combines two powerful soft decision decoding techniques which
were previously regarded in the literature as competitive, namely, the
Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation
based on adaptive parity check matrices, recently proposed by Jiang and
Narayanan. Building on the Jiang-Narayanan algorithm, we present a
belief-propagation based algorithm with a significant reduction in
computational complexity. We introduce the concept of using a
belief-propagation based decoder to enhance the soft-input information prior to
decoding with an algebraic soft-decision decoder. Our algorithm can also be
viewed as an interpolation multiplicity assignment scheme for algebraic
soft-decision decoding of Reed-Solomon codes.Comment: Submitted to IEEE for publication in Jan 200
Achievable Information Rates for Coded Modulation with Hard Decision Decoding for Coherent Fiber-Optic Systems
We analyze the achievable information rates (AIRs) for coded modulation
schemes with QAM constellations with both bit-wise and symbol-wise decoders,
corresponding to the case where a binary code is used in combination with a
higher-order modulation using the bit-interleaved coded modulation (BICM)
paradigm and to the case where a nonbinary code over a field matched to the
constellation size is used, respectively. In particular, we consider hard
decision decoding, which is the preferable option for fiber-optic communication
systems where decoding complexity is a concern. Recently, Liga \emph{et al.}
analyzed the AIRs for bit-wise and symbol-wise decoders considering what the
authors called \emph{hard decision decoder} which, however, exploits \emph{soft
information} of the transition probabilities of discrete-input discrete-output
channel resulting from the hard detection. As such, the complexity of the
decoder is essentially the same as the complexity of a soft decision decoder.
In this paper, we analyze instead the AIRs for the standard hard decision
decoder, commonly used in practice, where the decoding is based on the Hamming
distance metric. We show that if standard hard decision decoding is used,
bit-wise decoders yield significantly higher AIRs than symbol-wise decoders. As
a result, contrary to the conclusion by Liga \emph{et al.}, binary decoders
together with the BICM paradigm are preferable for spectrally-efficient
fiber-optic systems. We also design binary and nonbinary staircase codes and
show that, in agreement with the AIRs, binary codes yield better performance.Comment: Published in IEEE/OSA Journal of Lightwave Technology, 201
Iterative Soft Input Soft Output Decoding of Reed-Solomon Codes by Adapting the Parity Check Matrix
An iterative algorithm is presented for soft-input-soft-output (SISO)
decoding of Reed-Solomon (RS) codes. The proposed iterative algorithm uses the
sum product algorithm (SPA) in conjunction with a binary parity check matrix of
the RS code. The novelty is in reducing a submatrix of the binary parity check
matrix that corresponds to less reliable bits to a sparse nature before the SPA
is applied at each iteration. The proposed algorithm can be geometrically
interpreted as a two-stage gradient descent with an adaptive potential
function. This adaptive procedure is crucial to the convergence behavior of the
gradient descent algorithm and, therefore, significantly improves the
performance. Simulation results show that the proposed decoding algorithm and
its variations provide significant gain over hard decision decoding (HDD) and
compare favorably with other popular soft decision decoding methods.Comment: 10 pages, 10 figures, final version accepted by IEEE Trans. on
Information Theor
Construction of Capacity-Achieving Lattice Codes: Polar Lattices
In this paper, we propose a new class of lattices constructed from polar
codes, namely polar lattices, to achieve the capacity \frac{1}{2}\log(1+\SNR)
of the additive white Gaussian-noise (AWGN) channel. Our construction follows
the multilevel approach of Forney \textit{et al.}, where we construct a
capacity-achieving polar code on each level. The component polar codes are
shown to be naturally nested, thereby fulfilling the requirement of the
multilevel lattice construction. We prove that polar lattices are
\emph{AWGN-good}. Furthermore, using the technique of source polarization, we
propose discrete Gaussian shaping over the polar lattice to satisfy the power
constraint. Both the construction and shaping are explicit, and the overall
complexity of encoding and decoding is for any fixed target error
probability.Comment: full version of the paper to appear in IEEE Trans. Communication
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