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IEEE Transactions on Information Theory

Graphical Characterizations of Linear Programming Pseudocodewords for Cycle Codes

Nathan Axvig, Deanna Dreher

Published Jul 11, 2013• 8 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2013.2265693

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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.
Linear-programming pseudocodewords play a pivotal role in our understanding of the linear-programming decoding algorithms. These pseudocodewords are known to be equivalent to the graph-cover pseudocodewords. The latter pseudocodewords, when viewed as points in the multidimensional Euclidean space, lie inside a fundamental cone. This fundamental cone depends on the choice of a parity-check matrix of a code, rather than on the choice of the code itself. The cone does not depend on the channel, over which the code is employed. The knowledge of the boundaries of the fundamental cone could help in studying various properties of the pseudocodewords, such as their minimum pseudoweight, pseudoredundancy of the codes, etc. For the binary codes, the full characterization of the fundamental cone was derived by Koetter et al. However, if the underlying alphabet is large, such characterization becomes more involved. In this work, a characterization of the fundamental cone for codes over F3is discussed.
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.
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.
Characterization of Graph-cover Pseudocodewords of Codes over F3F_3F3​
Linear-programming pseudocodewords play a pivotal role in our understanding of the linear-programming decoding algorithms. These pseudocodewords are known to be equivalent to the graph-cover pseudocodewords. The latter pseudocodewords, when viewed as points in the multidimensional Euclidean space, lie inside a fundamental cone. This fundamental cone depends on the choice of a parity-check matrix of a code, rather than on the choice of the code itself. The cone does not depend on the channel, over which the code is employed. The knowledge of the boundaries of the fundamental cone could help in studying various properties of the pseudocodewords, such as their minimum pseudoweight, pseudoredundancy of the codes, etc. For the binary codes, the full characterization of the fundamental cone was derived by Koetter et al. However, if the underlying alphabet is large, such characterization becom is more involved. In this work, a characterization of the fundamental cone for codes over F3F_3F3​ is discussed.
Pseudocodewords, and in particular minimal pseudocodewords, play an important role in understanding the performance of linear programming (LP) decoding. In this paper, we investigate minimal pseudocodewords of binary Hamming codes described by full-rank parity-check matrices. We first provide some general results on minimal pseudocodewords with support size 3 of a binary parity-check matrix. We also prove a lower bound on the minimum binary symmetric channel (BSC) pseudoweight of a binary parity-check matrix. Then we prove that a full-rank parity-check matrix of a binary Hamming code has minimal pseudocodewords of certain types whose support sizes are larger than 3. Interestingly enough, the BSC pseudoweight of all these minimal pseudocodewords is 2. Using this fact as well as the above-mentioned lower bound, we further prove that a full-rank parity-check matrix of a binary Hamming code has minimum BSC pseudoweight 2. Moreover, the additive white Gaussian noise channel (AWGNC) pseudoweight of all these minimal pseudocodewords is 3. Based on numerical observations, we conjecture that a full-rank parity-check matrix of a binary Hamming code has minimum AWGNC pseudoweight 3. Finally, we provide more properties of a subset of minimal pseudocodewords of a full-rank parity-check matrix of a binary Hamming code.
Iterative decoding and linear programming decoding are guaranteed to converge to the maximum-likelihood codeword when the underlying Tanner graph is cycle-free. Therefore, cycles are usually seen as the culprit of low-density parity-check (LDPC) codes. In this paper, we argue in the context of graph cover pseudocodeword that, for a code that permits a cycle-free Tanner graph, cycles have no effect on error performance as long as they are a part of redundant rows. Specifically, we characterize all parity-check matrices that are pseudocodeword-free for such class of codes.