Construction of protograph-based LDPC codes with chordless short cycles

Controlling small size trapping sets and short cycles can result in LDPC codes with large minimum distance \(d_{\min}\). We prove that short cycles with a chord are the root of several trapping sets and eliminating these cycles increases \(d_{\min}\). We show that the lower bounds on \(d_{\min}\) of an LDPC code with chordless short cycles, girths 6 (and 8), and column weights \(\gamma\) (and 3), respectively, are \(2\gamma\) (and 10), which is a significant improvement compared to the existing bounds \(\gamma+1\) (and 6). Necessary and sufficient conditions for exponent matrices of protograph-based LDPC codes with chordless short cycles are proposed for any type of protographs, single-edge and multiple-edge, regular and irregular. The application of our method to girth-6 QC-LDPC codes shows that the removal of those cycles improves previous results in the literature.