Design of capacity-approaching irregular low-density parity-check codes

We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on the work of Richardson and...

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Veröffentlicht in:	IEEE transactions on information theory Jg. 47; H. 2; S. 619 - 637
Hauptverfasser:	Richardson, T.J., Shokrollahi, M.A., Urbanke, R.L.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York IEEE 01.02.2001 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:	Algorithms Asymptotic properties Binary system Codes Density Design engineering Error detection coding Graphs Messages Optimization Strategy Symmetry
ISSN:	0018-9448, 1557-9654
Online-Zugang:	Volltext
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Abstract	We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on the work of Richardson and Urbanke (see ibid., vol.47, no.2, p.599-618, 2000). Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds.
AbstractList	We design low-density parity-check (LDPQ codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on the work of Richardson and Urbanke (see ibid., vol.47, no.2, p.599-618, 2000). Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds. We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on the work of Richardson and Urbanke (see ibid., vol.47, no.2, p.599-618, 2000). Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds
Author	Urbanke, R.L. Shokrollahi, M.A. Richardson, T.J.
Author_xml	– sequence: 1 givenname: T.J. surname: Richardson fullname: Richardson, T.J. organization: Lucent Technol. Bell Labs., Murray Hill, NJ, USA – sequence: 2 givenname: M.A. surname: Shokrollahi fullname: Shokrollahi, M.A. – sequence: 3 givenname: R.L. surname: Urbanke fullname: Urbanke, R.L.
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References	ref13 Richardson (ref20) Luby (ref10) ref12 ref15 Horn (ref19) 1999 ref14 Price (ref23) 1997; 11 ref11 Shwartz (ref21) 1995 ref2 ref1 ref17 Billingsley (ref22) 1995 ref16 ref18 ref8 ref7 ref9 ref4 ref3 ref6 ref5 Luby (ref24) 2001; 47
References_xml	– volume-title: A proof of the stability condition for LDPC codes ident: ref20 – ident: ref5 doi: 10.1109/TIT.1981.1056404 – start-page: 364 volume-title: Proc. 9th Annu. ACM-SIAM Symp. Discrete Algorithms ident: ref10 article-title: Analysis of random processes via and-or tree evaluation – ident: ref7 doi: 10.1109/ISIT.1998.708706 – volume-title: Large Deviations for Performance Analysis year: 1995 ident: ref21 – ident: ref12 doi: 10.1109/ICC.1993.397441 – volume: 47 start-page: 569 year: 2001 ident: ref24 article-title: Practical loss-resilient codes publication-title: IEEE Trans. Inform. Theory doi: 10.1109/18.910575 – ident: ref4 doi: 10.1109/18.910579 – ident: ref11 doi: 10.1109/ISIT.2000.866295 – ident: ref15 doi: 10.1007/3-540-46796-3_7 – volume: 11 start-page: 341 year: 1997 ident: ref23 article-title: Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces publication-title: J. Global Optimiz. doi: 10.1023/A:1008202821328 – ident: ref8 doi: 10.1109/26.795809 – ident: ref2 doi: 10.1145/258533.258573 – ident: ref9 doi: 10.1007/978-3-7091-2945-6 – ident: ref14 doi: 10.1109/18.748992 – ident: ref16 doi: 10.1109/4234.905935 – ident: ref13 doi: 10.7551/mitpress/4347.001.0001 – ident: ref17 doi: 10.1007/978-1-4613-0165-3_9 – volume-title: Iterative decoding and pseudocodewords year: 1999 ident: ref19 – ident: ref1 doi: 10.1109/18.910577 – volume-title: Probability and Measure year: 1995 ident: ref22 – ident: ref18 doi: 10.1017/s0963548396002878 – ident: ref3 doi: 10.1109/18.556668 – ident: ref6 doi: 10.1145/276698.276756
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Snippet	We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular... We design low-density parity-check (LDPQ codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular...
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SubjectTerms	Algorithms Asymptotic properties Binary system Codes Density Design engineering Error detection coding Graphs Messages Optimization Strategy Symmetry
Title	Design of capacity-approaching irregular low-density parity-check codes
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