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arXiv:cs/0504013 · IEEE Transactions on Information Theory

Pseudocodewords of Tanner Graphs

Christine A. Kelley, Deepak Sridhara

Published Nov 1, 2007• 95 citations
arXivHTMLPDF

Abstract

This paper presents a detailed analysis of pseudocodewords of Tanner graphs. Pseudocodewords arising on the iterative decoder's computation tree are distinguished from pseudocodewords arising on finite degree lifts. Lower bounds on the minimum pseudocodeword weight are presented for the BEC, BSC, and AWGN channel. Some structural properties of pseudocodewords are examined, and pseudocodewords and graph properties that are potentially problematic with min-sum iterative decoding are identified. An upper bound on the minimum degree lift needed to realize a particular irreducible lift-realizable pseudocodeword is given in terms of its maximal component, and it is shown that all irreducible lift-realizable pseudocodewords have components upper bounded by a finite value$t$that is dependent on the graph structure. Examples and different Tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed. The results obtained provide some insights in relating the structure of the Tanner graph to the pseudocodeword distribution and suggest ways of designing Tanner graphs with good minimum pseudocodeword weight.

DOI
10.1109/tit.2007.907501

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IEEE Transactions on Information Theory•Aug 24, 2009•24 citations
This paper presents a detailed analysis of pseudocodewords of Tanner graphs. Pseudocodewords arising on the iterative decoder's computation tree are distinguished from pseudocodewords arising on finite degree lifts. Lower bounds on the minimum pseudocodeword weight are presented for the BEC, BSC, and AWGN channel. Some structural properties of pseudocodewords are examined, and pseudocodewords and graph properties that are potentially problematic with min-sum iterative decoding are identified. An upper bound on the minimum degree lift needed to realize a particular irreducible lift-realizable pseudocodeword is given in terms of its maximal component, and it is shown that all irreducible lift-realizable pseudocodewords have components upper bounded by a finite valuetttthat is dependent on the graph structure. Examples and different Tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed. The results obtained provide some insights in relating the structure of the Tanner graph to the pseudocodeword distribution and suggest ways of designing Tanner graphs with good minimum pseudocodeword weight.
Abstract—This paper presents a detailed analysis of pseudocodewords of Tanner graphs. Pseudocodewords arising on the iterative decoder’s computation tree are distinguished from pseudocodewords arising on finite degree lifts. Lower bounds on the minimum pseudocodeword weight are presented for the BEC, BSC, and AWGN channel. Some structural properties of pseudocodewords are examined, and pseudocodewords and graph properties that are potentially problematic with min-sum iterative decoding are identified. An upper bound on the minimum degree lift needed to realize a particular irreducible lift-realizable pseudocodeword is given in terms of its maximal component, and it is shown that all irreducible lift-realizable pseudocodewords have components upper bounded by a finite value t that is dependent on the graph structure. Examples and different Tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed. The results obtained provide some insights in relating the structure of the Tanner graph to the pseudocodeword distribution and suggest ways of designing Tanner graphs with good minimum pseudocodeword weight. Index Terms—Iterative decoding, low-density parity-check (LDPC) codes, min-sum iterative decoder, pseudocodewords. I.
The success of modern algorithms for the decoding problem such as message-passing iterative decoding and linear programming decoding lies in their local nature. This feature allows the algorithms to be extremely fast and capable of correcting more errors than guaranteed by the classical minimum distance of the code. Nonetheless, the performance of these decoders depends crucially on the Tanner graph representation of the code. In order to understand this choice of representation, we need to analyze the pseudocodewords of the Tanner graph of a code. These pseudocodewords are outputs of local decoding algorithms which may not be legitimate codewords. In this dissertation, we introduce a lifted fundamental cone and show that there is a one-to-one correspondence between graph cover pseudocodewords of a binary code and integer points in the lifted fundamental cone. We use this fact to prove the rationality of the generating function of the pseudocodewords for a general binary parity-check code. Our approach also yields algorithms for producing this generating function and provides tools for studying the irreducible pseudocodewords. Understanding irreducible pseudocodewords is crucial to determining the best representation of a code. Moreover, combining these techniques with the recent characterization of fundamental cone over F_3, we can analyze ternary parity-check codes. Finally, we make progress in the study of more general nonbinary codes by determining constraints satisfied by all pseudocodewords of a code over F_p where p is prime.
Since the invention of turbo codes, there has been an explosion of interest in iterative decoding and graphical representation of codes. This thesis examines the iterative decoding of codes defined on graphs with cycles, which appears to be an efficient means of achieving the Shannon limit. Much of this analysis is on the iterative min-sum decoding of tail-biting codes and cycle codes. We have identified the pseudocodeword as the cause of the suboptimal performance of the iterative decoder, and we have obtained a union bound for the performance of the iterative decoder on both AWGN and BSC channels. Using the union bound argument, for cycle codes, we have shown that the performance of the iterative decoder is asymptotically as good as that of the ML decoder. As for tail-biting codes, the same thing is true if the lowest weight pseudocodeword is at least the minimum weight of the code. Unfortunately, the analysis of tail-biting codes and cycle codes does not extend to turbo codes and low density parity check codes in general. Our next approach is to determine the average behavior of message passing algorithms by studying the evolution of their "message" densities. For the class of "repeat and accumulate" serially concatenated turbo-like codes, we have devised an algorithm for determining their "threshold" values. When the signal-to-noise ratio is larger than the threshold value, the error probabilities of message passing algorithms approach zero, whereas if the signal-to-noise ratio is less than the threshold value, the error probability stays bounded away from zero. Some message passing algorithms and graphical representation of codes are efficient means of devising ML or MAP decoding algorithms. We have proposed a junction tree representation for linear block codes, and we have shown that the minimum junction tree can be less complex than the minimal trellis.
The performance of linear programming decoding is determined by the set of nonzero linear programming pseudocodewords. The minimum pseudoweight of this set of linear programming pseudocodewords is also generally accepted as a good predictor of the performance of iterative message-passing decoding. Since the linear programming decoder has a natural description based on the Tanner graph of a code, linear programming pseudocodewords can be analyzed from a graphical viewpoint. In this paper, two graphical characterizations of linear programming pseudocodewords of cycle codes are provided: one for the entire set of linear programming pseudocodewords, and one for the set of minimal linear programming pseudocodewords. The first of these characterizations is used to determine a formula for the minimum degree of a graph cover necessary to realize a linear programming pseudocodeword of a cycle code, and the second is used to prove a result concerning the asymptotic performance of the linear programming decoder when considering transmission over either the additive white Gaussian noise channel or the binary symmetric channel. Finally, a discussion contrasting these two characterizations both with each other and with Horn's characterization of bad pseudocycles is provided.
The role of pseudocodewords in causing non-codeword outputs in linear programming decoding, graph cover decoding, and iterative message-passing decoding is investigated. The three main types of pseudocodewords in the literature-linear programming pseudocodewords, graph cover pseudocodewords, and computation tree pseudocodewords-are reviewed and connections between them are explored. Some discrepancies in the literature on minimal and irreducible pseudocodewords are highlighted and clarified, and the minimal degree cover necessary to realize a pseudocodeword is found. Additionally, some conditions for the existence of connected realizations of graph cover pseudocodewords are given. This allows for further analysis of when graph cover pseudocodewords induce computation tree pseudocodewords. Finally, an example is offered that shows that existing theories on the distinction between graph cover pseudocodewords and computation tree pseudocodewords are incomplete.