Local-optimality guarantees based on paths for optimal decoding

Guy Even, Nissim Halabi

Research output: Contribution to journalArticlepeer-review


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 repeataccumulate 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.

Original languageEnglish
Pages (from-to)2049-2063
Number of pages15
JournalSIAM Journal on Discrete Mathematics
Issue number4
StatePublished - 2013


  • Coding theory
  • Error probability
  • Iterative message-passing decoding
  • Linear programming decoding
  • Local optimality
  • Maximum likelihood
  • Repeat-accumulate codes
  • Tanner codes


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