## ADMM LP Decoding of Non-Binary LDPC Codes in <inline-formula> <tex-math notation="LaTeX">$ \mathbb {F}_{2^{m}}$ </tex-math></inline-formula>

- Xishuo Liu, Stark C. Draper
- IEEE Transactions on Information Theory
- 2016

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- Published 2005 in IEEE Transactions on Information Theory

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 d/sub frac/ 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 /spl lceil/d/sub frac//2/spl rceil/-1 errors and that there are codes with d/sub frac/=/spl Omega/(n/sup 1-/spl epsi//). 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.

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@article{Feldman2005UsingLP,
title={Using linear programming to Decode Binary linear codes},
author={Jon Feldman and Martin J. Wainwright and David R. Karger},
journal={IEEE Transactions on Information Theory},
year={2005},
volume={51},
pages={954-972}
}