In this paper, a new method is given for counting cycles in the Tanner graph of a (Type-I) quasi-cyclic (QC) low-density parity-check (LDPC) code which the complexity mainly is dependent on the base matrix, independent from the CPM-size of the constructed code. Interestingly, for large CPM-sizes, in comparison of the existing methods, this algorithm is the first approach which efficiently counts the cycles in the Tanner graphs of QC-LDPC codes. In fact, the algorithm recursively counts the cycles in the parity-check matrix column-by-column by finding all non-isomorph tailless backtrackless closed (TBC) walks in the base graph and enumerating theoretically their corresponding cycles in the same equivalent class. Moreover, this approach can be modified in few steps to find the cycle distributions of a class of LDPC codes based on Affine permutation matrices (APM-LDPC codes). Interestingly, unlike the existing methods which count the cycles up to $2g-2$, where $g$ is the girth, the proposed algorithm can be used to enumerate the cycles of arbitrary length in the Tanner graph. Moreover, the proposed cycle searching algorithm improves upon various previously known methods, in terms of computational complexity and memory requirements.

翻译：本文提出了一种在（I型）准循环低密度奇偶校验（QC-LDPC）码的Tanner图中计数环路的新方法，其复杂度主要依赖于基矩阵，而与所构造码的CPM大小无关。有趣的是，对于较大的CPM大小，相较于现有方法，该算法是首个能高效计数QC-LDPC码Tanner图中环路的方案。实际上，该算法通过寻找基图中所有非同构无尾回溯闭（TBC）游走，并从理论上枚举其同一等价类中的对应环路，从而按列递归地计数校验矩阵中的环路。此外，该方法可通过少量步骤修改，用于求出一类基于仿射置换矩阵的LDPC码（APM-LDPC码）的环路分布。有趣的是，与现有仅能计数长度至$2g-2$（其中$g$为围长）的环路的方法不同，所提算法可用于枚举Tanner图中任意长度的环路。同时，与多种先前已知方法相比，该环路搜索算法在计算复杂度和内存需求方面均有所改进。