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优异性能而受到广泛关注。基本捕获集是导致LDPC码错误平层现象的主要原因。我们研究了与捕获集相关的典型图结构，包括θ图、哑铃图以及带弦的短环。基于$θ(2,2,2)$、$θ(1,3,3)$和$D(4,4;0)$的Turán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，通过移除特定带弦短环结构，我们还可消除ETS。通过上述方法获得的ETS最小规模显著提升。为评估实际影响，我们分析了ETS的谱半径并构造了QC-LDPC码，展示了错误平层区的误帧率性能。