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.

翻译：众所周知，给定 \(b\ge 0\)，在二进制线性码中寻找具有最小 \(a\) 的 \((a,b)\)-陷阱集是NP难问题。本文证明，对于具有有界树宽的码，该问题可以相对于码长以线性复杂度求解。此外，假设已知与二进制线性码的树宽相对应的树分解，我们还提供了一种具体算法，以线性复杂度计算给定 \(b\) 下的最小 \(a\) 值以及相应 \((a, b)\)-陷阱集的数量。仿真实验验证了所提算法的正确性。