From Cages to Trapping Sets and Codewords: A Technique to Derive Tight Upper Bounds on the Minimum Size of Trapping Sets and Minimum Distance of LDPC Codes

TL;DR

This paper establishes a novel connection between cages in graph theory and trapping sets in LDPC codes, enabling the derivation of tight upper bounds on the smallest trapping sets and minimum codeword weight, which are crucial for understanding error floors.

Contribution

It introduces a method to derive tight upper bounds on trapping set sizes and minimum distance of LDPC codes using graph theory results on cages, linking two fields innovatively.

Findings

Tight upper bounds on trapping set sizes are derived for various LDPC codes.

The bounds often match known lower bounds, indicating their tightness.

Minimum codeword weight bounds are also established, improving code analysis.

Abstract

Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that non-zero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minimum distance, of variable-regular LDPC codes as a function of variable degree and girth.