On the lifting degree of girth-8 QC-LDPC codes

The lifting degree and the deterministic construction of quasi-cyclic low-density parity-check (QC-LDPC) codes have been extensively studied, with many construction methods in the literature, including those based on finite geometry, array-based codes, computer search, and combinatorial techniques. In this paper, we focus on the lifting degree $p$ required for achieving a girth of 8 in $(3,L)$ fully connected QC-LDPC codes, and we propose an improvement over the classical lower bound $p\ge 2L-1$, enhancing it to $p\ge \sqrt{5L^2-11L+\frac{13}{2}}+\frac{1}{2}$. Moreover, we demonstrate that for girth-8 QC-LDPC codes containing an arithmetic row in the exponent matrix, a necessary condition for achieving a girth of 8 is $p\ge \frac{1}{2}{L}^2+\frac{1}{2}L$. Additionally, we present a corresponding deterministic construction of $(3,L)$ QC-LDPC codes with girth 8 for any $p\ge \frac{1}{2}{L}^2+\frac{1}{2}L+\lfloor \frac{L-1}{2}\rfloor$, which approaches the lower bound of $\frac{1}{2}{L}^2+\frac{1}{2}L$. Under the same conditions, this construction achieves a smaller lifting degree compared to prior methods. To the best of our knowledge, the proposed order of lifting degree matches the smallest known, on the order of $\frac{1}{2}{L}^2+\mathcal{O}(L)$.