. We apply the technique of local-optimality to a family of Tanner codes. Inverse polynomial bounds in the code length are proved on the word error probability of LP-decoding for this family of Tanner codes.

DOI: 10.48550/arxiv.1203.1854

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Local-optimality guarantees for optimal decoding based on paths

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Local-Optimality Guarantees Based on Paths for Optimal Decoding

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This paper presents a unified analysis framework that captures recent advances in the study of local-optimality characterizations for codes on graphs. These local-optimality characterizations are based on combinatorial structures embedded in the Tanner graph of the code. Local optimality implies both unique maximum likelihood optimality and unique linear programming (LP) decoding optimality. Also, an iterative message-passing decoding algorithm is guaranteed to find the unique locally optimal codeword if one exists. We demonstrate an instance of this proof technique by considering a definition of local optimality that is based on the simplest combinatorial structures in Tanner graphs, namely, paths of length $h$ . We apply the technique of local optimality to binary Tanner codes (including any low-density parity-check code, and in particular any irregular repeat-accumulate code with both even and odd repetition factors). Inverse polynomial bounds in the code length are proved on the word error probability of LP decoding for binary Tanner codes. When the local codes are restricted to single parity-check codes, these bounds also hold for decoding by a certain iterative message-passing decoding algorithm.This paper presents a unified analysis framework that captures recent advances in the study of local-optimality characterizations for codes on graphs. These local-optimality characterizations are based on combinatorial structures embedded in the Tanner graph of the code. Local optimality implies both unique maximum likelihood optimality and unique linear programming (LP) decoding optimality. Also, an iterative message-passing decoding algorithm is guaranteed to find the unique locally optimal codeword if one exists. We demonstrate an instance of this proof technique by considering a definition of local optimality that is based on the simplest combinatorial structures in Tanner graphs, namely, paths of length $h$. We apply the technique of local optimality to binary Tanner codes (including any low-density parity-check code, and in particular any irregular repeat-accumulate code with both even and odd repetition factors). Inverse polynomial bounds in the code length are proved on the word error probability of LP decoding for binary Tanner codes. When the local codes are restricted to single parity-check codes, these bounds also hold for decoding by a certain iterative message-passing decoding algorithm.

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IEEE International Symposium on Information Theory (ISIT)Jul 1, 20122 citations

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Nissim Halabi, Guy Even

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On Decoding Irregular Tanner Codes With Local-Optimality Guarantees

Nissim Halabi, Guy Even

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IEEE Transactions on Information TheoryOct 10, 20136 citations

Local Optimality Certificates for LP Decoding of Tanner Codes

Nissim Halabi, Guy Even

We present a new combinatorial characterization for local optimality of a codeword in an irregular Tanner code. The main novelty in this characterization is that it is based on a linear combination of subtrees in the computation trees. These subtrees may have any degree in the local code nodes and may have any height (even greater than the girth). We expect this new characterization to lead to improvements in bounds for successful decoding. We prove that local optimality in this new characterization implies ML-optimality and LP-optimality, as one would expect. Finally, we show that is possible to compute efficiently a certificate for the local optimality of a codeword given an LLR vector.We present a new combinatorial characterization for local optimality of a codeword in an irregular Tanner code. The main novelty in this characterization is that it is based on a linear combination of subtrees in the computation trees. These subtrees may have any degree in the local code nodes and may have any height (even greater than the girth). We expect this new characterization to lead to improvements in bounds for successful decoding. We prove that local optimality in this new characterization implies ML-optimality and LP-optimality, as one would expect. Finally, we show that is possible to compute efficiently a certificate for the local optimality of a codeword given an LLR vector.

arXiv:1104.4646Apr 24, 20110 citations

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. We apply the technique of local optimality to binary Tanner codes (including any low-density parity-check code, and in particular any irregular repeat-accumulate code with both even and odd repetition factors). Inverse polynomial bounds in the code length are proved on the word error probability of LP decoding for binary Tanner codes. When the local codes are restricted to single parity-check codes, these bounds also hold for decoding by a certain iterative message-passing decoding algorithm.