Efficient implementation of linear programming decoding

This paper explores ideas for fast linear programming (LP) decoding of low-density parity-check (LDPC) codes. We first propose a modification of adaptive LP decoding, and prove that it performs LP decoding by solving a number of linear programs that contain at most one linear constraint derived from each of the parity-check constraints. Then, as a step toward designing an efficient LP solver that exploits the structure of LDPC codes, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithms is to solve a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a framework for designing preconditioners to be used with the iterative methods for solving these systems. The effectiveness of the proposed approach is demonstrated via both analytical and simulation results.