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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Vydáno v:IEEE transactions on information theory Ročník 47; číslo 2; s. 619 - 637
Hlavní autoři: Richardson, T.J., Shokrollahi, M.A., Urbanke, R.L.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.02.2001
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Algorithms
Asymptotic properties
Binary system
Codes
Density
Design engineering
Error detection coding
Graphs
Messages
Optimization
Strategy
Symmetry
ISSN:0018-9448, 1557-9654
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Shrnutí: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.
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ISSN:0018-9448
1557-9654
DOI:10.1109/18.910578