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 Turán numbers of $θ(2,2,2)$, $θ(1,3,3)$ and $D(4,4;0)$, we prove that any $(a,b)$-ETS with $g=8$ variable-regular $γ$ satisfies the inequality $b\geq aγ-\frac{a(\sqrt{24a-23}-1)}{4}$, 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.

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