The lifting degree and the deterministic construction of quasi-cyclic low-density parity-check (QC-LDPC) codes have been extensively studied due to their direct impact on the efficiency and error-correction performance of modern communication systems. 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 improve the classical lower bound \(p\ge 2L-1\) to \(p\ge \sqrt{5L^2-11L+\frac{13}{2}}+\frac{1}{2}\) . Moreover, for a QC-LDPC code containing an arithmetic row in its exponent matrix, we show that 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 a girth of 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 and exhibits better performance. The proposed lifting degree order matches the smallest known, on the order of \(\frac{1}{2}L^2+\mathcal {O} (L)\) .
