Construction of protograph-based LDPC codes with chordless short cycles
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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 γ (and 3), respectively, are 2γ (and 10), which is a significant improvement compared to the existing bounds γ+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.
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