Design of High-Rate LDPC Codes Based on Matroid Theory

In this letter, sufficient conditions for the determination of the girth are studied from the perspective of matroid theory. The girth of a Tanner graph is at least 2(t
₁
+ 2) if t
₁
specific conditions are simultaneously met. A novel method of constructing high-rate low-density parity-check (LDPC) codes is proposed based on the matroid theory. The parity-check matrices of the constructed LDPC codes are in the form of H = [I|H
₂
] with H
₂
constructed under the conditions of a given girth and a fixed column weight (e.g., W
_c
= 4 or W
_c
= 6). Simulation results verify that the proposed LDPC codes outperform those in the literature over additive white Gaussian noise channels in terms of bit error rate performance.