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.

翻译：LDPC码因其接近香农极限的卓越性能而受到广泛关注。基本陷阱集是导致LDPC码错误平层现象的主要原因。我们研究了与陷阱集相关的典型图结构，包括θ图、哑铃图以及带弦短环。基于$θ(2,2,2)$、$θ(1,3,3)$和$D(4,4;0)$的图兰数，我们证明：在Tanner图中任意两个8环不共享公共变量节点的条件下，任何围长$g=8$的变量正则$(a,b)$-ETS满足不等式$b\geq aγ-\frac{a(\sqrt{24a-23}-1)}{4}$。此外，通过移除某些带弦短环结构，还可进一步消除ETS。通过这些方法获得的ETS最小尺寸显著增大。为评估实际影响，我们分析了ETS的谱半径，并构造了QC-LDPC码以展示错误平层区域的误帧率。