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International Symposium on Turbo Codes and Iterative Information Processing

Asymptotic gaps between BP decoding and local-MAP decoding for low-density parity-check codes

Ryuhei Mori, Kenta Kasai, Tomoharu Shibuya, Kohichi Sakaniwa

Published Sep 1, 2008• 3 citations
Publisher

Abstract

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.

DOI
10.1109/turbocoding.2008.4658691

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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 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.
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) 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 binary memoryless symmetric channels using random elements from irregular low density parity check code (LDPC) ensembles, and belief propagation (BP) decoding.Under the assumption that the corresponding Tanner graph includes a non-vanishing fraction of degree 2 variable nodes, we determine the large blocklength behavior of the bit error rate for noise levels below threshold.More precisely, we show that the BP bit error rate is, asymptotically, F/n, where n is the blocklength and F an explicit constant.Surprisingly, this is the same behavior found for maximum likelihood (ML) decoding, implying that the BP and ML error floor are asymptotically equal.
In this correspondence, we first investigate some analytical aspects of the recently proposed improved decoding algorithm for low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We derive a necessary and sufficient condition for the improved decoding algorithm to successfully complete decoding when the decoder is initialized to guess a predetermined number of guesses after the standard message-passing terminates at a stopping set. Furthermore, we present improved bounds on the number of bits to be guessed for successful completion of the decoding process when a stopping set is encountered. Under suitable conditions, we derive a lower bound on the number of iterations to be performed for complete decoding of the stopping set. We then present a superior, novel improved decoding algorithm for LDPC codes over the binary erasure channel (BEC). The proposed algorithm combines the observation that a considerable fraction of unsatisfied check nodes in the neighborhood of a stopping set are of degree two, and the concept of guessing bits to perform simple and intuitive graph-theoretic manipulations on the Tanner graph. The proposed decoding algorithm has a complexity similar to previous improved decoding algorithms. Finally, we present simulation results of short-length codes over BEC that demonstrate the superiority of our algorithm over previous improved decoding algorithms for a wide range of bit error rates