Complexity analysis of interior point methods for LP decoding

Linear programming (LP) decoders can outperform currently used message-passing decoders in channel coding applications, but require prohibitively large complexity on even moderately sized codes. Previous works have proposed complexity-reducing algorithms that either relax the problem or modify the number of constraints; however, little work is done in optimizing solver implementation. We show that popular LP solvers like LIPSOL may not be efficient for LP decoding (LPD), and that an equivalent dual LP problem can be solved with equal accuracy but much more quickly. We propose an improved primal-dual method (iPD-MPC) whose overall runtime for both problem formulations outpreform LIPSOL. Additionally, as an alternative for memory-limited systems, we propose an improved hybrid gradient descent and Newton's method (iGD-NM) that further decreases overall runtime. In this way, we make LPD more feasible for channel codes of practical lengths.