Matrix-Theoretic Design of Cycle-Aware Quasi-Cyclic LDPC Codes

Abstract

Quasi-cyclic low-density parity-check (QC-LDPC) codes are widely adopted in modern communication standards due to their near-capacity performance and hardware-friendly structures. In this paper, we propose a matrix-theoretic, cycleaware framework for constructing QC-LDPC codes from arbitrary base matrices by systematically optimizing their shift values to maximize the girth and minimize the number of cycles with length girth. We present a cycle-aware iterative algorithm that identifies critical edges on the largest number of shortest cycles and modifies their shift values to break multiple cycles simultaneously. The framework supports flexible lifting sizes, which enable a single base matrix to generate codes with arbitrary lengths. Constructive bounds show that girth-6 and girth-8 codes can be achieved with lifting sizes as small as 4 and 14, respectively. Simulation results show that the proposed codes achieve good performance under iterative decoding while retaining the quasicyclic structure essential for hardware implementation.