Linear Complexity Computation of Code Distance and Minimum Size of Trapping Sets for LDPC Codes with Bounded Treewidth

TL;DR

The authors tackle the NP-hard MINIMUM TRAPPING SET problem for binary LDPC codes by exploiting bounded treewidth. They develop a dynamic-programming framework over rooted nice tree decompositions to compute the minimum trap size $a$ for a given $b$ and to count the number of such trapping sets, achieving linear time in the code length $n$ with complexity depending on the treewidth $k$. The approach extends to general $b$ by augmenting the DP with additional state variables, yielding a tractable method for enumerating smallest $(a,b)$-trapping sets in codes of bounded treewidth. Simulations on spatially coupled LDPC codes demonstrate accuracy and substantial speedups over exhaustive search, validating the method’s practicality for long codes. Overall, the paper provides a principled, scalable technique to quantify and characterize trapping structures in LDPC codes under a bounded-treewidth assumption, with direct implications for understanding and mitigating error floors.

Abstract

It is well known that, given \(b\ge 0\), finding an $(a,b)$-trapping set with the minimum \(a\) in a binary linear code is NP-hard. In this paper, we demonstrate that this problem can be solved with linear complexity with respect to the code length for codes with bounded treewidth. Furthermore, suppose a tree decomposition corresponding to the treewidth of the binary linear code is known. In that case, we also provide a specific algorithm to compute the minimum \(a\) and the number of the corresponding \((a, b)\)-trapping sets for a given \(b\) with linear complexity. Simulation experiments are presented to verify the correctness of the proposed algorithm.

Figures (4)

• Figure 1: We depict a bipartite graph $G$ with $tw(G) = 2$, along with its two tree decompositions. Different tree decompositions may have different widths, but we are primarily concerned with the tree decomposition of minimum width.
• Figure 2: The rooted nice tree decomposition corresponding to the bipartite graph in Fig.\ref{['a']}. This nice tree decomposition can be obtained by adding some "auxiliary nodes" (dashed lines) to the tree decomposition in Fig.\ref{['b']}.
• Figure 3: A Venn diagram illustrating the symmetric difference and intersection of sets $R_2$ and $R_3$.
• Figure 4: A randomly generated spatially coupled LDPC code. The variable nodes have a regular degree of 3. The black regions indicate positions in the parity-check matrix where the entries are 1, while other positions are 0.

Theorems & Definitions (16)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Definition 4
• Corollary 2
• proof
• Corollary 3
• proof