Theoretical Bounds for the Size of Elementary Trapping Sets by Graphic Methods

TL;DR

This paper derives precise bounds on elementary trapping set sizes in Tanner graphs for LDPC codes, providing theoretical insights that help in designing codes with improved error floor performance.

Contribution

It introduces exact Turán numbers for specific graph structures and establishes new bounds on ETSs, aiding in LDPC code design to mitigate error floors.

Findings

Derived the Turán number of θ(1,2,2) for Tanner graphs.

Proved bounds on ETS sizes for girth 6 and 8 Tanner graphs.

Designed LDPC codes with lower error floors demonstrating improved performance.

Abstract

Elementary trapping sets (ETSs) are the main culprits for the performance of LDPC codes in the error floor region. Due to the large quantity, complex structures, and computational difficulties of ETSs, how to eliminate dominant ETSs in designing LDPC codes becomes a pivotal issue to improve the error floor behavior. In practice, researchers commonly address this problem by avoiding some special graph structures to free specific ETSs in Tanner graph. In this paper, we deduce the accurate Tur\'an number of $\theta(1,2,2)$ and prove that all $(a,b)$-ETSs in Tanner graph with variable-regular degree $d_L(v)=\gamma$ must satisfy the bound $b\geq a\gamma-\frac{1}{2}a^2$, which improves the lower bound obtained by Amirzade when the girth is 6. For the case of girth 8, by limiting the relation between any two 8-cycles in the Tanner graph, we prove a similar inequality $b\geq a\gamma-\frac{a(\sqrt{8a-7}-1)}{2}$. The simulation results show that the designed codes have good performance with lower error floor over additive white Gaussian noise channels.