Exhausting Error-Prone Patterns in LDPC Codes

TL;DR

This work proves that exhaustively identifying error-prone patterns in LDPC codes is NP-complete but provides efficient algorithms for short codes, enabling better code analysis and optimization, especially for low-error applications.

Contribution

It introduces a tree-based exhaustive search algorithm for identifying stopping and trapping sets in LDPC codes of practical lengths, with proven bounds and improved efficiency.

Findings

Efficient exhaustive determination of small stopping and trapping sets in LDPC codes.

Provides tight upper bounds on BER and FER for arbitrary codes over BECs.

Demonstrates the algorithm's superiority through extensive numerical experiments.

Abstract

It is proved in this work that exhaustively determining bad patterns in arbitrary, finite low-density parity-check (LDPC) codes, including stopping sets for binary erasure channels (BECs) and trapping sets (also known as near-codewords) for general memoryless symmetric channels, is an NP-complete problem, and efficient algorithms are provided for codes of practical short lengths n~=500. By exploiting the sparse connectivity of LDPC codes, the stopping sets of size <=13 and the trapping sets of size <=11 can be efficiently exhaustively determined for the first time, and the resulting exhaustive list is of great importance for code analysis and finite code optimization. The featured tree-based narrowing search distinguishes this algorithm from existing ones for which inexhaustive methods are employed. One important byproduct is a pair of upper bounds on the bit-error rate (BER) & frame-error rate (FER) iterative decoding performance of arbitrary codes over BECs that can be evaluated for any value of the erasure probability, including both the waterfall and the error floor regions. The tightness of these upper bounds and the exhaustion capability of the proposed algorithm are proved when combining an optimal leaf-finding module with the tree-based search. These upper bounds also provide a worst-case-performance guarantee which is crucial to optimizing LDPC codes for extremely low error rate applications, e.g., optical/satellite communications. Extensive numerical experiments are conducted that include both randomly and algebraically constructed LDPC codes, the results of which demonstrate the superior efficiency of the exhaustion algorithm and its significant value for finite length code optimization.