On the lifting degree of girth-8 QC-LDPC codes
Haoran Xiong, Guanghui Wang, Zhiming Ma, Guiying Yan
TL;DR
The paper tackles the problem of minimizing the lifting degree $p$ needed to achieve girth $8$ in $(3,L)$ QC-LDPC codes. It derives a new universal lower bound $p \ge \sqrt{5L^2-11L+\frac{13}{2}}+\frac{1}{2}$ and, under an arithmetic second row of the exponent matrix, a tighter bound $p \ge \frac{1}{2}L^2+\frac{1}{2}L$, then provides deterministic constructions that realize girth $8$ with $p$ close to these bounds. Specifically, a construction achieves $p \ge \frac{1}{2}L^2+\frac{1}{2}L+\lfloor \frac{L-1}{2}\rfloor$, approaching the lower bound and outperforming prior methods for the same arithmetic-row condition. The results yield shorter QC-LDPC codes with strong girth properties and improved practical performance, extending the known $\mathcal{O}(L^2)$ scaling to near-optimal constants for this class of codes.
Abstract
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 $p\geq 2L-1$, enhancing it to $p\geq \sqrt{5L^2-11L+\frac{13}{2}}+\frac{1}{2}$. 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 $p\geq \frac{1}{2}L^2+\frac{1}{2}L$. Additionally, we present a corresponding deterministic construction of $(3,L)$ QC-LDPC codes with girth 8 for any $p\geq \frac{1}{2}L^2+\frac{1}{2}L+\lfloor \frac{L-1}{2}\rfloor$, 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 $\frac{1}{2}L^2+\mathcal{O} (L)$.