Iterative Approximate Linear Programming Decoding of LDPC Codes With Linear Complexity

• Published in: IEEE transactions on information theory Vol. 55; no. 11; pp. 4835 - 4859
• Main Author: Burshtein, D.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.11.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Approximation; Belief propagation; Codes; Coding, codes; Complexity; Computation; Computational complexity; Decoding; Error analysis; Error correction; Exact sciences and technology; Information theory; Information, signal and communications theory; Iterative algorithms; Iterative decoding; Linear approximation; Linear programming; Linear programming decoding; low-density parity-check (LDPC) codes; Mathematical analysis; Mathematical models; Parity check codes; Processor scheduling; Scheduling algorithm; Signal and communications theory; Telecommunications and information theory; Vectors (mathematics); Iterative method; Linear programming; Scheduling; Algorithm; Computational complexity; Iterative decoding; Linear programming decoding; Credal approach; low-density parity-check (LDPC) codes; Linear complexity; Convergence rate; Parity check codes; Error correcting code; Error correction; Objective function; Approximate solution
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2030477

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.