Effects of Single-Cycle Structure on Iterative Decoding for Low-Density Parity-Check Codes

TL;DR

This paper analyzes how single-cycle structures in LDPC codes affect the accuracy of iterative decoding over the BEC, deriving formulas to evaluate their impact on error probability for finite blocklengths.

Contribution

It introduces recursive formulas to evaluate the impact of single-cycle structures on bit error probability in LDPC codes with finite blocklengths.

Findings

The dominant error difference scales as 1/n and is influenced by cycle structures.

The derived formulas accurately predict error probabilities even for small blocklengths.

Single-cycle structures significantly affect decoding performance in finite-length LDPC codes.

Abstract

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For fixed numbers of BP iterations, the bit error probability approaches a limit as blocklength tends to infinity, and the limit is obtained via density evolution. On the other hand, the difference between the bit error probability of codes with blocklength $n$ and that in the large blocklength limit is asymptotically $\alpha(\epsilon,t)/n + \Theta(n^{-2})$ where $\alpha(\epsilon,t)$ denotes a specific constant determined by the code ensemble considered, the number $t$ of iterations, and the erasure probability $\epsilon$ of the BEC. In this paper, we derive a set of recursive formulas which allows evaluation of the constant $\alpha(\epsilon,t)$ for standard irregular ensembles. The dominant difference $\alpha(\epsilon,t)/n$ can be considered as effects of cycle-free and single-cycle structures of local graphs. Furthermore, it is confirmed via numerical simulations that estimation of the bit error probability using $\alpha(\epsilon,t)$ is accurate even for small blocklengths.