## Using linear programming to decode binary linear codes (2005)

Venue: | IEEE TRANS. INFORM. THEORY |

Citations: | 113 - 11 self |

### BibTeX

@ARTICLE{Feldman05usinglinear,

author = {Jon Feldman and Martin J. Wainwright and David R. Karger},

title = {Using linear programming to decode binary linear codes},

journal = {IEEE TRANS. INFORM. THEORY},

year = {2005},

volume = {51},

number = {3},

pages = {954--972}

}

### Years of Citing Articles

### OpenURL

### Abstract

A new method is given for performing approximate maximum-likelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor graph or parity-check representation of the code. The resulting “LP decoder” generalizes our previous work on turbo-like codes. A precise combinatorial characterization of when the LP decoder succeeds is provided, based on pseudocodewords associated with the factor graph. Our definition of a pseudocodeword unifies other such notions known for iterative algorithms, including “stopping sets, ” “irreducible closed walks, ” “trellis cycles, ” “deviation sets, ” and “graph covers.” The fractional distance ��— ™ of a code is introduced, which is a lower bound on the classical distance. It is shown that the efficient LP decoder will correct up to ��— ™ P I errors and that there are codes with ��— ™ a @ I A. An efficient algorithm to compute the fractional distance is presented. Experimental evidence shows a similar performance on low-density parity-check (LDPC) codes between LP decoding and the min-sum and sum-product algorithms. Methods for tightening the LP relaxation to improve performance are also provided.