Upper Bounding the Performance of Arbitrary Finite LDPC Codes on Binary Erasure Channels

TL;DR

This paper introduces a new tree-based method for upper bounding the bit error rates of finite LDPC codes on BECs, providing tight bounds across all erasure probabilities including the error floor region.

Contribution

A novel tree-based technique for upper bounding BERs of finite LDPC codes on BECs, applicable to all operating regions and capable of tight bounds with optimal leaf-finding modules.

Findings

Upper bounds are tight in asymptotic order.

Method can exhaustively analyze small stopping sets.

Provides deterministic error floor guarantees.

Abstract

Assuming iterative decoding for binary erasure channels (BECs), a novel tree-based technique for upper bounding the bit error rates (BERs) of arbitrary, finite low-density parity-check (LDPC) codes is provided and the resulting bound can be evaluated for all operating erasure probabilities, including both the waterfall and the error floor regions. This upper bound can also be viewed as a narrowing search of stopping sets, which is an approach different from the stopping set enumeration used for lower bounding the error floor. When combined with optimal leaf-finding modules, this upper bound is guaranteed to be tight in terms of the asymptotic order. The Boolean framework proposed herein further admits a composite search for even tighter results. For comparison, a refinement of the algorithm is capable of exhausting all stopping sets of size <14 for irregular LDPC codes of length n=500, which requires approximately 1.67*10^25 trials if a brute force approach is taken. These experiments indicate that this upper bound can be used both as an analytical tool and as a deterministic worst-performance (error floor) guarantee, the latter of which is crucial to optimizing LDPC codes for extremely low BER applications, e.g., optical/satellite communications.