Elementary trapping sets (ETSs) are the main culprits for the performance of LDPC codes in the error floor region. Due to the large quantity, complex structures, and computational difficulties of ETSs, how to eliminate dominant ETSs in designing LDPC codes becomes a pivotal issue to improve the error floor behavior. In practice, researchers commonly address this problem by avoiding some special graph structures to free specific ETSs in Tanner graph. In this paper, we deduce the accurate Tur\'an number of $\theta(1,2,2)$ and prove that all $(a,b)$-ETSs in Tanner graph with variable-regular degree $d_L(v)=\gamma$ must satisfy the bound $b\geq a\gamma-\frac{1}{2}a^2$, which improves the lower bound obtained by Amirzade when the girth is 6. For the case of girth 8, by limiting the relation between any two 8-cycles in the Tanner graph, we prove a similar inequality $b\geq a\gamma-\frac{a(\sqrt{8a-7}-1)}{2}$. The simulation results show that the designed codes have good performance with lower error floor over additive white Gaussian noise channels.

翻译：基本陷阱集是LDPC码在错误平层区域性能的主要影响因素。由于基本陷阱集数量庞大、结构复杂且计算困难，如何在设计LDPC码时消除主要基本陷阱集成为改善错误平层行为的关键问题。实践中，研究者通常通过避免某些特殊图结构来排除Tanner图中的特定基本陷阱集。本文推导了$\theta(1,2,2)$的精确Turán数，并证明了在变量正则度为$d_L(v)=\gamma$的Tanner图中，所有$(a,b)$-基本陷阱集必须满足界限$b\geq a\gamma-\frac{1}{2}a^2$，该结果在围长为6时改进了Amirzade得到的下界。针对围长为8的情况，通过限制Tanner图中任意两个8-环的关系，我们证明了类似不等式$b\geq a\gamma-\frac{a(\sqrt{8a-7}-1)}{2}$。仿真结果表明，在加性高斯白噪声信道上，所设计的码字具有更低的错误平层和良好性能。