On Trapping Sets and Guaranteed Error Correction Capability of LDPC Codes and GLDPC Codes

TL;DR

This paper explores the relationship between girth, trapping sets, and error correction in LDPC and GLDPC codes, providing bounds and conditions for guaranteed correction capabilities using bit flipping algorithms.

Contribution

It introduces bounds on trapping set sizes and expands the understanding of error correction limits for LDPC and GLDPC codes under bit flipping decoding.

Findings

Lower bounds on variable node set expansion based on Moore bound.

Upper bounds on guaranteed error correction capability via trapping set analysis.

Generalized LDPC codes can correct a linear fraction of errors with good expanders.

Abstract

The relation between the girth and the guaranteed error correction capability of $\gamma$-left regular LDPC codes when decoded using the bit flipping (serial and parallel) algorithms is investigated. A lower bound on the size of variable node sets which expand by a factor of at least $3 \gamma/4$ is found based on the Moore bound. An upper bound on the guaranteed error correction capability is established by studying the sizes of smallest possible trapping sets. The results are extended to generalized LDPC codes. It is shown that generalized LDPC codes can correct a linear fraction of errors under the parallel bit flipping algorithm when the underlying Tanner graph is a good expander. It is also shown that the bound cannot be improved when $\gamma$ is even by studying a class of trapping sets. A lower bound on the size of variable node sets which have the required expansion is established.