Loop Calculus Helps to Improve Belief Propagation and Linear Programming Decodings of Low-Density-Parity-Check Codes

TL;DR

This paper demonstrates how loop calculus can enhance belief propagation and linear programming decodings of LDPC codes by identifying and correcting critical loops, thereby reducing error floors in noisy communication channels.

Contribution

The paper introduces a method to incorporate loop calculus into decoding algorithms, improving their performance by addressing problematic loops in the graphical representation of codes.

Findings

Loop calculus improves LP decoding accuracy.

The method corrects dangerous pseudo-codeword configurations.

Error floor is significantly reduced in tested codes.

Abstract

We illustrate the utility of the recently developed loop calculus for improving the Belief Propagation (BP) algorithm. If the algorithm that minimizes the Bethe free energy fails we modify the free energy by accounting for a critical loop in a graphical representation of the code. The log-likelihood specific critical loop is found by means of the loop calculus. The general method is tested using an example of the Linear Programming (LP) decoding, that can be viewed as a special limit of the BP decoding. Considering the (155,64,20) code that performs over Additive-White-Gaussian-Noise channel we show that the loop calculus improves the LP decoding and corrects all previously found dangerous configurations of log-likelihoods related to pseudo-codewords with low effective distance, thus reducing the code's error-floor.