Tur\'{a}n-Theoretic Bounds on Several Elementary Trapping Sets in LDPC Codes

TL;DR

This paper establishes Turán-theoretic bounds on elementary trapping sets in LDPC codes, analyzes their spectral properties, and demonstrates how to eliminate them to improve error floor performance.

Contribution

It introduces Turán number-based bounds on trapping sets, proposes methods to eliminate certain ETSs, and evaluates their impact on LDPC code performance.

Findings

Derived bounds on elementary trapping sets using Turán numbers.

Proposed elimination of ETSs by removing specific short-cycle structures.

Constructed QC-LDPC codes demonstrating improved error floor performance.

Abstract

LDPC codes have attracted significant attention because of their superior performance close to the Shannon limit. Elementary trapping sets are the main cause of the error floor phenomenon in LDPC codes. We consider typical graphs related to trapping sets, including theta graphs, dumbbell graphs, and short cycles with chords. Based on the Tur\'{a}n numbers of $\theta(2,2,2)$, $\theta(1,3,3)$ and $D(4,4;0)$, we prove that any $(a,b)$-ETS with $g=8$ variable-regular $\gamma$ satisfies the inequality $b\geq a\gamma-\frac{a(\sqrt{24a-23}-1)}{4}$, provided that any two 8-cycles in the Tanner graph do not share common variable node. In addition, we can also eliminate ETSs by removing certain short-cycle structures with chords. The minimum sizes of ETSs obtained through these methods are significantly increased. To assess practical impact , we analyze spectral radii of the ETSs and construct QC-LDPC codes to show frame error rates in the error floor region.