Error Correction Capability of Column-Weight-Three LDPC Codes

TL;DR

This paper analyzes the error correction limits of column-weight-three LDPC codes with Gallager A decoding, establishing conditions on Tanner graph cycles and showing fundamental correction bounds for large code lengths.

Contribution

It proves necessary cycle length conditions for correcting multiple errors and demonstrates that no large ensemble of these codes can correct a linear fraction of errors.

Findings

Cycle length constraints are necessary for correcting k errors.

Large ensembles cannot correct a linear fraction of errors.

Error correction capability is fundamentally limited by Tanner graph properties.

Abstract

In this paper, we investigate the error correction capability of column-weight-three LDPC codes when decoded using the Gallager A algorithm. We prove that the necessary condition for a code to correct $k \geq 5$ errors is to avoid cycles of length up to $2k$ in its Tanner graph. As a consequence of this result, we show that given any $\alpha>0, \exists N $ such that $\forall n>N$, no code in the ensemble of column-weight-three codes can correct all $\alpha n$ or fewer errors. We extend these results to the bit flipping algorithm.