The capacity of low-density parity-check codes under message-passing decoding

We present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any binary-input memoryless channel with discrete or continuous output alphabets. Transmitting at rates below this capacity, a randomly chosen element of th...

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Bibliographic Details
Published in:	IEEE transactions on information theory Vol. 47; no. 2; pp. 599 - 618
Main Authors:	Richardson, T.J., Urbanke, R.L.
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.02.2001 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Algorithms Alphabets Binary system Channels Codes Decoders Decoding Error detection coding Errors Linear programming Transmission Turbo codes
ISSN:	0018-9448, 1557-9654
Online Access:	Get full text
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Abstract	We present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any binary-input memoryless channel with discrete or continuous output alphabets. Transmitting at rates below this capacity, a randomly chosen element of the given ensemble will achieve an arbitrarily small target probability of error with a probability that approaches one exponentially fast in the length of the code. (By concatenating with an appropriate outer code one can achieve a probability of error that approaches zero exponentially fast in the length of the code with arbitrarily small loss in rate.) Conversely, transmitting at rates above this capacity the probability of error is bounded away from zero by a strictly positive constant which is independent of the length of the code and of the number of iterations performed. Our results are based on the observation that the concentration of the performance of the decoder around its average performance, as observed by Luby et al. in the case of a binary-symmetric channel and a binary message-passing algorithm, is a general phenomenon. For the particularly important case of belief-propagation decoders, we provide an effective algorithm to determine the corresponding capacity to any desired degree of accuracy. The ideas presented in this paper are broadly applicable and extensions of the general method to low-density parity-check codes over larger alphabets, turbo codes, and other concatenated coding schemes are outlined.
AbstractList	In this paper, we present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any binary-input memoryless channel with discrete or continuous output alphabets. Transmitting at rates below this capacity, a randomly chosen element of the given ensemble will achieve an arbitrarily small target probability of error with a probability that approaches one exponentially fast in the length of the code. We present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any binary-input memoryless channel with discrete or continuous output alphabets. Transmitting at rates below this capacity, a randomly chosen element of the given ensemble will achieve an arbitrarily small target probability of error with a probability that approaches one exponentially fast in the length of the code. (By concatenating with an appropriate outer code one can achieve a probability of error that approaches zero exponentially fast in the length of the code with arbitrarily small loss in rate.) Conversely, transmitting at rates above this capacity the probability of error is bounded away from zero by a strictly positive constant which is independent of the length of the code and of the number of iterations performed. Our results are based on the observation that the concentration of the performance of the decoder around its average performance, as observed by Luby et al. in the case of a binary-symmetric channel and a binary message-passing algorithm, is a general phenomenon. For the particularly important case of belief-propagation decoders, we provide an effective algorithm to determine the corresponding capacity to any desired degree of accuracy. The ideas presented in this paper are broadly applicable and extensions of the general method to low-density parity-check codes over larger alphabets, turbo codes, and other concatenated coding schemes are outlined. We present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any binary-input memoryless channel with discrete or continuous output alphabets. Transmitting at rates below this capacity, a randomly chosen element of the given ensemble will achieve an arbitrarily small target probability of error with a probability that approaches one exponentially fast in the length of the code. (By concatenating with an appropriate outer code one can achieve a probability of error that approaches zero exponentially fast in the length of the code with arbitrarily small loss in rate.) Conversely, transmitting at rates above this capacity the probability of error is bounded away from zero by a strictly positive constant which is independent of the length of the code and of the number of iterations performed. Our results are based on the observation that the concentration of the performance of the decoder around its average performance, as observed by Luby et al. in the case of a binary-symmetric channel and a binary message-passing algorithm, is a general phenomenon. For the particularly important case of belief-propagation decoders, we provide an effective algorithm to determine the corresponding capacity to any desired degree of accuracy. The ideas presented in this paper are broadly applicable and extensions of the general method to low-density parity-check codes over larger alphabets, turbo codes, and other concatenated coding schemes are outlined
Author	Urbanke, R.L. Richardson, T.J.
Author_xml	– sequence: 1 givenname: T.J. surname: Richardson fullname: Richardson, T.J. organization: Lucent Technols. Bell Labs., Murray Hill, NJ, USA – sequence: 2 givenname: R.L. surname: Urbanke fullname: Urbanke, R.L.
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References	ref13 ref12 ref20 ref11 ref10 ref2 Zyablov (ref6) 1975; 11 ref16 ref19 ref18 ref8 Gelblum (ref17) ref7 ref9 ref4 ref3 Luby (ref1) ref5 Mitzenmacher (ref14) 1998 Pearl (ref15) 1988
References_xml	– ident: ref8 doi: 10.1007/BF02579283 – volume: 11 start-page: 23 year: 1975 ident: ref6 article-title: Estimation of the error-correction complexity of Gallager low-density codes publication-title: Probl. Pered. Inform. – ident: ref9 doi: 10.1109/26.795809 – ident: ref2 doi: 10.1109/ICC.1993.397441 – ident: ref18 doi: 10.1109/isit.2000.866615 – ident: ref13 doi: 10.1109/TIT.2004.833352 – ident: ref20 doi: 10.1002/0471722154 – ident: ref4 doi: 10.1109/18.556667 – volume-title: A note on low density parity check codes for erasures and errors year: 1998 ident: ref14 – volume-title: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference year: 1988 ident: ref15 – ident: ref12 doi: 10.1002/0471200611 – ident: ref3 doi: 10.1109/18.748992 – ident: ref5 doi: 10.7551/mitpress/4347.001.0001 – ident: ref7 doi: 10.1109/TIT.1981.1056404 – ident: ref11 doi: 10.1109/4234.681360 – start-page: 271 volume-title: Proc. Int. Symp. Turbo Codes and Related Topics ident: ref17 article-title: Understanding serially concatenated codes from a support tree approach – ident: ref16 doi: 10.1017/CBO9780511814075 – ident: ref19 doi: 10.1109/glocom.1989.64230 – ident: ref10 doi: 10.1109/18.910578 – volume-title: Analysis of low density codes and improved designs using irregular graphs ident: ref1 doi: 10.1145/276698.276756
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Snippet	We present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over any... In this paper, we present a general method for determining the capacity of low-density parity-check (LDPC) codes under message-passing decoding when used over...
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SubjectTerms	Algorithms Alphabets Binary system Channels Codes Decoders Decoding Error detection coding Errors Linear programming Transmission Turbo codes
Title	The capacity of low-density parity-check codes under message-passing decoding
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