The Impact of the Distance Between Cycles on Elementary Trapping Sets

TL;DR

The paper investigates how the distance between cycles in LDPC Tanner graphs affects elementary trapping sets (ETSs), using theta graphs (negative distance) and dumbbell graphs (non-negative distance) to model ETS structures. By deriving Turán numbers for these graphs and applying a linear state-space model with spectral analysis, it shows that longer cycles and larger distances between cycles reduce the spectral radius $\rho(A_{sys})$ and thus the harmfulness of ETSs, aligning with ETS-elimination intuition. It also demonstrates that removing smaller-distance structures eliminates more ETSs, providing a theoretical basis for designing LDPC codes with improved error floors. A practical PEG-CYCLE algorithm is introduced to construct Tanner graphs with maximized cycle separation, and numerical results for QC-LDPC codes show competitive or superior performance compared to state-of-the-art methods. Overall, the work connects extremal graph theory and spectral theory to practical LDPC code design, offering both theoretical guarantees and actionable construction techniques.

Abstract

Elementary trapping sets (ETSs) are the main culprits of the performance of low-density parity-check (LDPC) codes in the error floor region. Due to their large quantities and complex structures, ETSs are difficult to analyze. This paper studies the impact of the distance between cycles on ETSs, focusing on two special graph classes: theta graphs and dumbbell graphs, which correspond to cycles with negative and non-negative distances, respectively. We determine the Turán numbers of these graphs and prove that increasing the distance between cycles can eliminate more ETSs. Additionally, using the linear state-space model and spectral theory, we prove that increasing the length of cycles or distance between cycles decreases the spectral radius of the system matrix, thereby reducing the harmfulness of ETSs. This is consistent with the conclusion obtained using Turán numbers. For specific cases when removing two 6-cycles with distance of -1, 0 and 1, respectively, we calculate the sizes, spectral radii, and error probabilities of ETSs. These results confirm that the performance of LDPC codes improves as the distance between cycles increases. Furthermore, we design the PEG-CYCLE algorithm, which greedily maximizes the distance between cycles in the Tanner graph. Numerical results show that the QC-LDPC codes constructed by our method achieve performance comparable to or even superior to state-of-the-art construction methods.

Figures (8)

• Figure 1: Figure $(a)$ is the theta graph $\theta(l_1,l_2,l_3)$. Figures $(b)$ and $(c)$ show the dumbbell graphs $D_b(r_1,r_2;q)$ for $q\geq 1$ and $q=0$, respectively.
• Figure 2: Figure $(a)$ labels each directed edge in the $(4,2)$-ETS, where circles represent variable nodes, squares represent check nodes, and black solid squares represent 1-degree check nodes. Figure $(b)$ illustrates the update relationships of the directed edges in this ETS, where each node corresponds to a directed edge from Figure $(a)$, and the directed edges indicate the message-passing process: the direction from the tail to the head signifies that the directed edge at the head participates in the update of the directed edge at the tail. Figure $(c)$ shows the corresponding VN graph.
• Figure 3: The variation of the spectral radius with respect to different cycle lengths and distances between cycles.
• Figure 4: The average error probability of ETSs in $\mathcal{S}_{\theta(1,2,2)-free}(10,4)$, $\mathcal{S}_{D_b(3,3;0)-free}(10,4)$, and $\mathcal{S}_{D_b(3,3;1)-free}(10,4)$ as the number of iterations increases for $\gamma=4$.
• Figure 5: An expansion with variable node $v_0$ as the root, where circles represent variable nodes and squares represent check nodes.

Theorems & Definitions (28)

• Definition 1
• Definition 2
• Remark 1
• Definition 3
• Definition 4
• Theorem 1: Mantel's theorem, mantel1907problem
• Theorem 2: Perron-Frobenius Theorem brouwer2011spectra
• Example 1
• Theorem 3: xiong2024theoretical
• Lemma 1