Characterizations of pseudo-codewords of (low-density) parity-check codes

Koetter, R.; Li, Wen-Ching Winnie; Vontobel, Pascal O.; Walker, Judy L.

doi:10.1016/j.aim.2006.12.010

Cited by 50 publications

(63 citation statements)

References 10 publications

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“…With this terminology, irreducible codewords coincide precisely with minimal codewords. Additionally, if p is irreducible as a codeword, then p is also irreducible as a pseudocodeword because if p were the sum of pseudocodewords then each of those pseudocodewords must consist of only zeros and ones and thus must be codewords themselves [11], which contradicts the irreducibility of p. Hence a vector p is an irreducible codeword if and only if it is a minimal codeword, if and only if it is a trivial irreducible pseudocodeword.…”

Section: Remark 43mentioning

confidence: 88%

“…One characterization of graph cover pseudocodewords is given by Koetter, Li, Vontobel, and Walker [11]: a vector p of nonnegative integers is an unscaled graph cover pseudocodeword if and only if it reduces modulo 2 to a codeword and it lies within the fundamental cone K ⊆ R n where…”

Section: Connections Between Lp and ML Decodingmentioning

confidence: 99%

“…If (1, 1, 1, 1) = x + y with x and y being nonzero nonnegative integer vectors, then x must have at least one coordinate which is 0 and one coordinate which is 1, and hence is not a pseudocodeword since it will not reduce modulo 2 to a codeword [11]. This means that (1, 1, 1, 1) is an irreducible pseudocodeword, but it is not a minimal pseudocodeword, since it does not lie on an edge of the fundamental cone.…”

Section: Remark 43mentioning

confidence: 99%

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Analysis of Connections Between Pseudocodewords

Axvig

Dreher

Morrison

et al. 2009

IEEE Trans. Inform. Theory

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Abstract-The role of pseudocodewords in causing noncodeword outputs in linear programming (LP) 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 a value for the minimal degree cover necessary to realize an LP 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.

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Section: Remark 43mentioning

confidence: 88%

Section: Connections Between Lp and ML Decodingmentioning

confidence: 99%

Section: Remark 43mentioning

confidence: 99%

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Analysis of Connections Between Pseudocodewords

Axvig

Dreher

Morrison

et al. 2009

IEEE Trans. Inform. Theory

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“…The set of codewords is the set of all binary assignments to the variable nodes such that at each check node, the modulo two sum of the variable node assignments connected to that check node is zero. The notion of covering graphs (or lifts of graphs) enters into the analysis of the graphbased iterative decoder in the explanation of pseudocodewords [11], [14], [17], [25].…”

Section: Preliminariesmentioning

confidence: 99%

LDPC codes from voltage graphs

Kelley

Walker

2008

2008 IEEE International Symposium on Information Theory

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Abstract-Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of abelianinevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained.

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“…Certain such restrictions on the weight distributions of the rows can only be satisfied if the parity-check matrix of the code has a sufficiently large number of judiciously chosen rows. Thus, recent work focused on introducing redundant rows into parity-check matrices of a code in order to ensure that the size of their smallest stopping sets are sufficiently large or equal to the minimum distance of the code [4], [5], [6], [7]. Since adding redundant rows to the parity-check matrix increases the decoding complexity of the code, it is important to understand the inherent trade-off between the size of the smallest stopping set and the number of rows in a parity-check matrix.…”

Section: Introductionmentioning

confidence: 99%

The Trapping Redundancy of Linear Block Codes

Laendner

Hehn

Milenković

et al. 2009

IEEE Trans. Inform. Theory

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We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in any parity-check matrix of a given code and the size of its smallest trapping set. Trapping sets with certain parameter sizes are known to cause error-floors in the performance curves of iterative belief propagation decoders, and it is therefore important to identify decoding matrices that avoid such sets. Bounds on the trapping redundancy are obtained using probabilistic and constructive methods, and the analysis covers both general and elementary trapping sets. Numerical values for these bounds are computed for the [2640, 1320] Margulis code and the class of projective geometry codes, and compared with some new code-specific trapping set size estimates.

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Characterizations of pseudo-codewords of (low-density) parity-check codes

Cited by 50 publications

References 10 publications

Analysis of Connections Between Pseudocodewords

Analysis of Connections Between Pseudocodewords

LDPC codes from voltage graphs

The Trapping Redundancy of Linear Block Codes

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