Turán-Theoretic Bounds on Several Elementary Trapping Sets in LDPC Codes

Abstract

LDPC codes have attracted significant attention due to their capacity-approaching performance. Elementary trapping sets are the main cause of the error floor phenomenon in LDPC codes. We investigate several graph structures associated with trapping sets, including theta graphs, dumbbell graphs, and short cycles with chords. Based on the Turán numbers of $\theta (2,2,2)$, $\theta (1,3,3)$ and $D(4,4;0)$, we prove that any $(a,b)$-ETS in a variable-regular Tanner graph with girth $g=8$ and variable degree $\gamma$ satisfies the inequality $b\ge a\gamma -\frac{a(\sqrt{24a-23}-1)}{4}$, 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 lower bounds on the minimum size of ETSs through these methods are improved. 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.