# Low-Density Parity-Check Decoding Algorithm Based on Symmetric Alternating Direction Method of Multipliers

## TL;DR

This paper introduces a new decoding algorithm for LDPC codes using symmetric ADMM, which improves decoding performance and reduces errors, especially at low SNR.

## Contribution

The novel contribution is the development of an LDPC decoding algorithm based on symmetric ADMM with improved efficiency and accuracy.

## Key findings

- The symmetric ADMM (S-ADMM) decoder outperforms conventional ADMM penalized decoders in frame error rate performance.
- The proposed algorithm demonstrates significant improvements in decoding performance for both standard LDPC and 5G codes.
- The S-ADMM algorithm satisfies contraction properties, ensuring stable and efficient iterative decoding.

## Abstract

The Alternating Direction Method of Multipliers (ADMM) has proven to be an efficient approach for implementing linear programming (LP) decoding of low-density parity-check (LDPC) codes. By introducing penalty terms into the LP decoding model’s objective function, ADMM-based variable node penalized decoding effectively mitigates non-integral solutions, thereby improving frame error rate (FER) performance, especially in the low signal-to-noise ratio (SNR) region. In this paper, we leverage the ADMM framework to derive explicit iterative steps for solving the LP decoding problem for LDPC codes with penalty functions. To further enhance decoding efficiency and accuracy, We propose an LDPC code decoding algorithm based on the symmetric ADMM (S-ADMM). We also establish some contraction properties satisfied by the iterative sequence of the algorithm. Through simulation experiments, we evaluate the proposed S-ADMM decoder using three standard LDPC codes and three representative fifth-generation (5G) codes. The results show that the S-ADMM decoder consistently outperforms conventional ADMM penalized decoders, offering significant improvements in decoding performance.