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IEEE Information Theory Workshop (ITW)

Minimum Pseudo-Codewords of LDPC Codes

Shu‐Tao Xia, Fang‐Wei Fu

Published Oct 1, 2006• 7 citations
Publisher

Abstract

In this paper, we study the minimum pseudo-weight and minimum pseudo-codewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelly, Sridhara, Xu and Rosenthal on the pseudo-weight of a pseudo-codeword of an LDPC code with girth greater than 4 is tight if and only if this pseudo-codeword is a real multiple of a codeword. Then, we show that the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is also a lower bound on the pseudo-weight of a pseudo-codeword of this LDPC code with girth 4, and this lower bound is tight if and only if this pseudo-codeword is a real multiple of a codeword. Using these results we further show that for some LDPC codes, there are no other minimum pseudo-codewords except the real multiples of minimum codewords. This means that LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximum-likelihood (ML) decoding approaches to 1 as the signal-to-noise ratio leads to infinity. Finally, some LDPC codes are listed to illustrate these results

DOI
10.1109/itw2.2006.323767

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In this correspondence, we study the minimum pseudoweight and minimum pseudocodewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelley, Sridhara, Xu, and Rosenthal on the pseudoweight of a nonzero pseudocodeword of an LDPC code whose Tanner graph has girth greater than is tight if and only if this pseudocodeword is a real multiple of a codeword. Then, the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is proved to be also a lower bound on the pseudoweight of a nonzero pseudocodeword of an LDPC code whose Tanner graph has girth , and this lower bound is tight if and only if this pseudocodeword is a real multiple of a codeword. Using these results we further obtain that for some LDPC codes, there are no other minimum pseudocodewords except the real multiples of minimum weight codewords. This means that the LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximum-likelihood decoding approaches as the signal-to-noise ratio (SNR) tends to infinity. Finally, some LDPC codes are listed to illustrate these results.

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Iterative message-passing decoders for low-density parity-check (LDPC) block codes are known to be subject to decoding failures due to so-called pseudo-codewords. These failures can cause the large signal-to-noise ratio performance of message-passing decoding to be worse than that predicted by the maximum-likelihood decoding union bound. In this paper we study the pseudo-codeword problem for the class of LDPC convolutional codes decoded continuously using an iterative, sliding window, message-passing decoder. In particular, for an LDPC convolutional code derived by unwrapping a quasi-cyclic LDPC block code, we show that the free pseudo-weight of the convolutional code is at least as large as the minimum pseudo-weight of the underlying quasi-cyclic code. This result parallels the well-known relationship between the free Hamming distance of convolutional codes and the minimum Hamming distance of their quasi-cyclic counterparts. Finally, simulation results are included that show improved performance for unwrapped LDPC convolutional codes compared to their underlying quasi-cyclic codes

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2006•16 citations
In this paper, we study the minimum pseudo-weight and minimum pseudo-codewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelly, Sridhara, Xu and Rosenthal on the pseudo-weight of a pseudo-codeword of an LDPC code with girth greater than 4 is tight if and only if this pseudo-codeword is a real multiple of a codeword. Then, we show that the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is also a lower bound on the pseudo-weight of a pseudo-codeword of this LDPC code with girth 4, and this lower bound is tight if and only if this pseudo-codeword is a real multiple of a codeword. Using these results we further show that for some LDPC codes, there are no other minimum pseudo-codewords except the real multiples of minimum codewords. This means that LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximum-likelihood (ML) decoding approaches to 1 as the signal-to-noise ratio leads to infinity. Finally, some LDPC codes are listed to illustrate these results
In this correspondence, we study the minimum pseudoweight and minimum pseudocodewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelley, Sridhara, Xu, and Rosenthal on the pseudoweight of a nonzero pseudocodeword of an LDPC code whose Tanner graph has girth greater than is tight if and only if this pseudocodeword is a real multiple of a codeword. Then, the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is proved to be also a lower bound on the pseudoweight of a nonzero pseudocodeword of an LDPC code whose Tanner graph has girth , and this lower bound is tight if and only if this pseudocodeword is a real multiple of a codeword. Using these results we further obtain that for some LDPC codes, there are no other minimum pseudocodewords except the real multiples of minimum weight codewords. This means that the LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximum-likelihood decoding approaches as the signal-to-noise ratio (SNR) tends to infinity. Finally, some LDPC codes are listed to illustrate these results.
In this letter, we study the minimum pseudo-codewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. We show that a lower bound of Chaichanavong and Siegel on the pseudo-weight of a pseudo-codeword is tight if and only if this pseudo-codeword is a real multiple of a codeword. Using this result we further show that for some LDPC codes, e.g., Euclidean plane and projective plane LDPC codes, there are no other minimum pseudo-codewords except the real multiples of minimum codewords.
Abstract — Considering the Linear Programming (LP) decoding of Low-Density-Parity-Check (LDPC) code [1], we describe an efficient algorithm for finding pseudo codewords with the smallest effective distance. The algorithm, coined LP-loop, starts from choosing randomly initial configuration of the noise. The configuration is modified through a discrete number of steps. Each step consists of two sub-steps. First, one applies LP decoder to the initial noise-configuration deriving a pseudo-codeword. Second, one finds configuration of the noise equidistant from the pseudo codeword and the all zeros codeword. The resulting noise configuration is used as an entry for the next step of the LP-loop algorithm. The iterations converge fast to a pseudo codeword neighboring the all zeros codeword — domain of the pseudo codeword attraction shares a common piece of error-surface with the all zeros codeword domain. Repeated many times the LP-loop procedure is characterized by the distribution function (frequency spectrum) of the pseudo codeword effective distance. Effective distance of the coding scheme is approximated by the shortest distance pseudo-codeword in the spectrum. Efficiency of the LP-loop algorithm is demonstrated on examples of the Tanner (155, 64, 20) code and Margulis p = 7 and p = 11 codes (672 and 2640 bits long respectively) performing over Additive-White-Gaussian-Noise (AWGN) channel.
Belief Propagation (BP) and Linear Programming (LP) decodings of Low Density Parity Check (LDPC) codes are discussed. We summarize results of instanton/pseudo-codeword approach developed for analysis of the error-floor domain of the codes. Instantons are special, code and decoding specific, configurations of the channel noise contributing most to the Frame-Error-Rate (FER). Instantons are decoded into pseudo-codewords. Instanton/pseudo-codeword with the lowest weight describes the largest Signal-to-Noise-Ratio (SNR) asymptotic of FER, while the whole spectra of the low weight instantons is descriptive of the FER vs SNR profile in the extended error-floor domain. First, we describe a general optimization method that allows to find the instantons for any coding/decoding. Second, we introduce LP-specific pseudo-codeword search algorithm that allows efficient calculations of the pseudo-codeword spectra. Finally, we discuss results of combined BP/LP error-floor exploration experiments for two model codes.
Belief propagation (BP) and linear programming (LP) decodings of low density parity check (LDPC) codes are discussed. We summarize results of instanton/pseudo-codeword approach developed for analysis of the error-floor domain of the codes. Instantons are special, code and decoding specific, configurations of the channel noise contributing most to the Frame-Error-Rate (FER). Instantons are decoded into pseudo- codewords. Instanton/pseudo-codeword with the lowest weight describes the largest Signal-to-Noise-Ratio (SNR) asymptotic of FER, while the whole spectra of the low weight instantons is descriptive of the FER vs SNR profile in the extended error-floor domain. First, we describe a general optimization method that allows to find the instantons for any coding/decoding. Second, we introduce LP-specific pseudo-codeword search algorithm that allows efficient calculations of the pseudo-codeword spectra. Finally, we discuss results of combined BP/LP error-floor exploration experiments for two model codes.
Iterative message-passing decoders for low-density parity-check (LDPC) block codes are known to be subject to decoding failures due to so-called pseudo-codewords. These failures can cause the large signal-to-noise ratio performance of message-passing decoding to be worse than that predicted by the maximum-likelihood decoding union bound. In this paper we study the pseudo-codeword problem for the class of LDPC convolutional codes decoded continuously using an iterative, sliding window, message-passing decoder. In particular, for an LDPC convolutional code derived by unwrapping a quasi-cyclic LDPC block code, we show that the free pseudo-weight of the convolutional code is at least as large as the minimum pseudo-weight of the underlying quasi-cyclic code. This result parallels the well-known relationship between the free Hamming distance of convolutional codes and the minimum Hamming distance of their quasi-cyclic counterparts. Finally, simulation results are included that show improved performance for unwrapped LDPC convolutional codes compared to their underlying quasi-cyclic codes