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 $\vert$ 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.