Programming Decoding Of Low-density Parity-check Codes Research Articles

On the Decomposition Method for Linear Programming Decoding of LDPC Codes

In this paper, we focus on solving the linear programming (LP) problem that arises in the decoding of low-density parity-check (LDPC) codes by means of the revised simplex method. In order to take advantage of the structure of the LP problem, we reformulate the dual LP and apply the idea of Dantzig-Wolfe (D-W) decomposition method to solve the problem. Each subproblem in the D-W decomposition method is an LP over a convex polyhedral cone. We define the convex polyhedral cone as local parity-check cone and discuss its special structures. Then, we enumerate its extreme rays and use these extreme rays to design an efficient method for the general LP decoding problem. The proposed method exhibits advantages in reducing both the storage and computational requirements.

Iterative Approximate Linear Programming Decoding of LDPC Codes With Linear Complexity

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.