  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.
