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

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.