Efficient implementation of linear programming decoding

TL;DR

This paper introduces a fast and efficient approach for linear programming decoding of LDPC codes by reducing the problem size, exploiting sparsity, and using preconditioning techniques to improve computational performance.

Contribution

It modifies adaptive LP decoding to remove unnecessary constraints and develops a sparse interior-point method with a novel preconditioning algorithm for faster decoding.

Findings

LP decoding can be performed with at most one constraint per parity check.

The proposed preconditioning algorithm improves the efficiency of solving linear systems.

Simulation results demonstrate the effectiveness of the fast LP decoding approach.

Abstract

While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem, and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. We first prove, by modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, that LP decoding can be performed by solving a number of LP problems that contain at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, 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 algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate iterative solution of such systems. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.