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arXiv:1010.0333 · IEEE Transactions on Information Theory

Effects of Single-Cycle Structure on Iterative Decoding of Low-Density Parity-Check Codes

Ryuhei Mori, Toshiyuki Tanaka, Kenta Kasai, Kohichi Sakaniwa

Published Sep 15, 2012• 2 citations
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Abstract

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For fixed numbers of BP iterations, the bit error probability approaches a limit as the blocklength tends to infinity, and the limit is obtained via density evolution. The finite-blocklength correction behaves like α(ε,t)/n+Θ(n-2) as the blocklengthntends to infinity where α(ε,t) denotes a specific constant determined by the code ensemble considered, the numbertof iterations, and the erasure probability ε of the BEC. In this paper, we derive a set of recursive formulas which allows the evaluation of the constant α(ε,t) for standard irregular ensembles. The dominant difference α(ε,t)/ncan be considered as effects of cycle-free and single-cycle structures of local graphs. Furthermore, it is confirmed via numerical simulations that estimation of the bit error probability using α(ε,t) is accurate even for small blocklengths.

DOI
10.1109/tit.2012.2216252

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We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For fixed numbers of BP iterations, the bit error probability approaches a limit as the blocklength tends to infinity, and the limit is obtained via density evolution. The finite-blocklength correction behaves like α(ε,t)/n+Θ(n-2) as the blocklengthntends to infinity where α(ε,t) denotes a specific constant determined by the code ensemble considered, the numbertof iterations, and the erasure probability ε of the BEC. In this paper, we derive a set of recursive formulas which allows the evaluation of the constant α(ε,t) for standard irregular ensembles. The dominant difference α(ε,t)/ncan be considered as effects of cycle-free and single-cycle structures of local graphs. Furthermore, it is confirmed via numerical simulations that estimation of the bit error probability using α(ε,t) is accurate even for small blocklengths.
Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength n tends to infinity. The difference between the bit error probability with blocklength n and the large-blocklength limit behaves asymptotically like ¿/n, where the coefficient ¿ depends on the ensemble, the number of iterations and the erasure probability of the BEC. In, ¿ is calculated for regular ensembles. In this paper, ¿ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of ¿ to the analytic result is significantly fast for irregular unexpurgated ensembles.
We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. The bit error probability for infinite block length is known by density evolution and it is well known that a difference between the bit error probability at finite iteration number for finite block length nnn and for infinite block length is asymptotically α/nα/nα/n, where ααα is a specific constant depending on the degree distribution, the iteration number and the erasure probability. Our main result is to derive an efficient algorithm for calculating ααα for regular ensembles. The approximation using ααα is accurate for (2,r)(2,r)(2,r)-regular ensembles even in small block length.
We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. Furthermore, a gap between the bit error probability after finite number of iterations for finite block length n and that for infinite block length is asymptotically α/n, where α denotes a speci..c constant determined by a degree distribution, a number of iterations and erasure probability. Our main result is to derive an ef..cient algorithm for calculating α for regular ensembles.
In this paper, we consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes. We consider MAP decoding not using a whole Tanner graph but only a neighborhood graph of fixed depth referred to as local-MAP decoding for deriving lower bounds of the error probability under message-passing decoding and bit-flipping decoding. The main result of this paper is to derive an asymptotic performance for regular ensembles under local-MAP decoding and to derive an asymptotic gap of the bit error probability between belief propagation (BP) and local-MAP decoding for irregular ensembles. Finally, we show the limit of the scaling parameter of these decodings when number of iterations tends to infinity.
We investigate the average erasure probability of the belief propagation algorithm over the binary erasure channel (BEC) for various finite-length low-density parity-check (LDPC) ensembles. In particular, we give tight upper bounds on the "error floor", i.e., on the contribution to the erasure probability stemming from relatively small deficiencies in the graph. We also define and analyse "expurgated" ensembles and give upper bounds on the error floor of "typical" codes. Finally, we show that typical codes of properly chosen ensembles have an erasure correcting radius which grows linearly in the block length. The presented method provides valuable insight into how finite-length codes should be designed. Although the standard ensemble can achieve capacity on the BEC, the construction of good finite-length codes requires careful control of the structure of the degree two variable nodes. In this paper, we investigate the standard ensemble, the Poisson ensemble, an ensemble in which degree two nodes are added in a cycle, and an ensemble in which different degree two variable nodes have distinct neighboring check nodes.
In this paper we present a new algorithm, denoted as TEP, to decode low-density parity-check (LDPC) codes over the Binary Erasure Channel (BEC). The TEP decoder is derived applying the expectation propagation (EP) algorithm with a tree- structured approximation. Expectation Propagation (EP) is a generalization to Belief Propagation (BP) in two ways. First, it can be used with any exponential family distribution over the cliques in the graph. Second, it can impose additional constraints on the marginal distributions. We use this second property to impose pair-wise marginal constraints in some check nodes of the LDPC code's Tanner graph. The algorithm has the same computational complexity than BP, but it can decode a higher fraction of errors when applied over the BEC. In this paper, we focus on the asymptotic performance of the TEP decoder, as the block size tends to infinity. We describe the TEP decoder by a set of differential equations that represents the residual graph evolution during the decoding process. The solution of these equations yields the capacity of this decoder for a given LDPC ensemble over the BEC. We show that the achieved capacity with the TEP is higher than the BP capacity, at the same computational complexity.