# Asymptotic Average Multiplicity of Structures within Different Categories of Trapping Sets, Absorbing Sets and Stopping Sets in Random Regular and Irregular LDPC Code Ensembles

## TL;DR

This paper analyzes the asymptotic average counts of various trapping set structures in random LDPC code ensembles, revealing their growth behavior and providing cycle-based estimates that are accurate even for finite-length codes.

## Contribution

It characterizes the asymptotic behavior of different trapping set types in LDPC ensembles and offers cycle-based estimates for structures with a single cycle.

## Key findings

- Average number of structures tends to infinity, a positive constant, or zero depending on cycle count.
- Cycle-based estimates are accurate for finite-length codes.
- Growth behavior depends on the structure's cycle content.

## Abstract

The performance of low-density parity-check (LDPC) codes in the error floor region is closely related to some combinatorial structures of the code's Tanner graph, collectively referred to as {\it trapping sets (TSs)}. In this paper, we study the asymptotic average number of different types of trapping sets such as {\em elementary TSs (ETS)}, {\em leafless ETSs (LETS)}, {\em absorbing sets (ABS)}, {\em elementary ABSs (EABS)}, and {\em stopping sets (SS)}, in random variable-regular and irregular LDPC code ensembles. We demonstrate that, regardless of the type of the TS, as the code's length tends to infinity, the average number of a given structure tends to infinity, to a positive constant, or to zero, if the structure contains no cycle, only one cycle, or more than one cycle, respectively. For the case where the structure contains a single cycle, we obtain an estimate of the expected number of the structure through the available approximations for the average number of its constituent cycle. These estimates, which are independent of the block length and only depend on the code's degree distributions, are shown to be accurate even for finite-length codes.