TY - JOUR

T1 - Iterative approximate linear programming decoding of LDPC codes with linear complexity

AU - Burshtein, David

N1 - Funding Information:
Manuscript received April 01, 2008; revised January 02, 2009. Current version published October 21, 2998. This work was supported by the Israel Science Foundation under Grant 927/05. The material in this paper was presented in part at the IEEE International Symposium on Information Theory (ISIT), Toronto, ON, Canada, July 2008 and at Turbo Coding 2008, Lausanne, Switzerland, September 2008. The author is with the School of Electrical Engineering, Tel-Aviv University, Tel-Aviv, Ramat–Aviv 69978, Israel (e-mail: [email protected]). Communicated by I. Sason, Associate Editor for Coding Theory. Color versions of Figures 1–5 in this paper are available online at http://iee-explore.ieee.org. Digital Object Identifier 10.1109/TIT.2009.2030477

PY - 2009

Y1 - 2009

N2 - The problem of low complexity linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered. An iterative algorithm, similar to min-sum and belief propagation, for efficient approximate solution of this problem was proposed by Vontobel and Koetter. In this paper, the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose. In particular, we are interested in obtaining a feasible vector in the LP decoding problem that is close to optimal in the following sense. The distance, normalized by the block length, between the minimum and the objective function value of this approximate solution can be made arbitrarily small. It is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length. Combined with previous results that have shown that the LP decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity. This is achieved by first applying the iterative LP decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits, and then correcting the remaining errors using some standard method. These conclusions are also extended to generalized LDPC codes.

AB - The problem of low complexity linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered. An iterative algorithm, similar to min-sum and belief propagation, for efficient approximate solution of this problem was proposed by Vontobel and Koetter. In this paper, the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose. In particular, we are interested in obtaining a feasible vector in the LP decoding problem that is close to optimal in the following sense. The distance, normalized by the block length, between the minimum and the objective function value of this approximate solution can be made arbitrarily small. It is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length. Combined with previous results that have shown that the LP decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity. This is achieved by first applying the iterative LP decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits, and then correcting the remaining errors using some standard method. These conclusions are also extended to generalized LDPC codes.

KW - Iterative decoding

KW - Linear programming decoding

KW - Low-density parity-check (LDPC) codes

UR - http://www.scopus.com/inward/record.url?scp=70350741443&partnerID=8YFLogxK

U2 - 10.1109/TIT.2009.2030477

DO - 10.1109/TIT.2009.2030477

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AN - SCOPUS:70350741443

SN - 0018-9448

VL - 55

SP - 4835

EP - 4859

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 11

ER -