An Efficient Strategy to Count Cycles in the Tanner Graph of Quasi-Cyclic LDPC Codes

In this paper, we present an efficient strategy to enumerate the number of $k$ -cycles, $g\leq k < 2g$ , in the Tanner graph of a quasi-cyclic low-density parity-check (QC-LDPC) code with girth $g$ using its polynomial parity-check matrix $H$ . This strategy works for both $(d_{v},d_{c})$ -regular and irregular QC-LDPC codes. In this approach, we note that the $m$ th power of the polynomial adjacency matrix can be used to describe walks of length $m$ in the protograph and can therefore be sufficiently described by the matrices $B_{m}(H) \triangleq (HH^{\mathsf {T}})^{\lfloor {m/2}\rfloor }H^{(m\mod 2)}$ , where $m\geq 0$ . We provide formulas for the number of $k$ -cycles, $\mathcal {N}_{k}$ , by just taking into account repetitions in some multisets constructed from the matrices $B_{m}(H)$ . This approach is shown to have low complexity. For example, in the case of QC-LDPC codes based on the $3\times n_{v}$ fully-connected protograph, the complexity of determining $\mathcal {N}_{k}$ , for $k=4,6,8,10$ and 12, is $O(n_{v}^{2}\log (N))$ , $O(n_{v}^{2}\log (n_{v})\log (N))$ , $O(n_{v}^{4}\log ^{4}(n_{v})\log (N))$ , $O(n_{v}^{4}\log (n_{v})\log (N))$ and $O(n_{v}^{6}\log ^{6}(n_{v})\log (N))$ , respectively. The complexity, depending logarithmically on the lifting factor $N$ , gives our approach, to the best of our knowledge, a significant advantage over previous works on the cycle distribution of QC-LDPC codes.