Theoretical Bounds for the Size of Elementary Trapping Sets by Graphic Methods

TL;DR

This work tackles LDPC error floors by bounding the size of elementary trapping sets (ETS) via Turán-number analysis of theta graphs in VN representations of variable-regular Tanner graphs. It proves $ex(n,\theta(1,2,2))=\left\lfloor n^2/4\right\rfloor$, leading to the girth-6 bound $b \ge a\gamma - \frac{1}{2}a^2$, and establishes a girth-8 bound $b \ge a\gamma - \frac{a(\sqrt{8a-7}-1)}{2}$ under a no-two-8-cycle-sharing condition, using $ex(a,\{C_3,\theta(2,2,2)\})$. By applying these results to LDPC code design, the paper shows how eliminating key theta-structures reduces small ETSs, and it demonstrates practical gains via QC-LDPC constructions with improved error-floor performance on AWGN channels. The constructions and simulations indicate the approach's effectiveness and broad applicability to regular QC-LDPC codes.

Abstract

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 Turán number of $θ(1,2,2)$ and prove that all $(a,b)$-ETSs in Tanner graph with variable-regular degree $d_L(v)=γ$ must satisfy the bound $b\geq aγ-\frac{1}{2}a^2$, 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 $b\geq aγ-\frac{a(\sqrt{8a-7}-1)}{2}$. The simulation results show that the designed codes have good performance with lower error floor over additive white Gaussian noise channels.

Figures (3)

• Figure 1: Figures (a) and (b) are two 6-cycles sharing one common check node and its VN graph $\theta(1,2,2)$, respectively. Figures (c) and (d) are two 8-cycles sharing two common check nodes and its VN graph $\theta(2,2,2)$, respectively. In these figures, circles are variable nodes and squares are check nodes.
• Figure 2: Figures (a), (b), (c) and (d) are the VN graphs of (8,0), (9,1), (8,2) and (7,3)-ETSs with girth 8 and $\gamma=3$, respectively.
• Figure 3: FER performance of $C_1$, $C_2$ and their counterparts

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4: The Turán number of $\theta(1,2,2)$
• Theorem 5: zhai2021turan
• Theorem 6: The Turán number of $\theta(2,2,2)$ and $C_3$
• Theorem 7
• Theorem 8
• Corollary 9
• Corollary 10