Protograph-based LDPC codes with chordless short cycles and large minimum distance

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, girth 6 and column weights gamma is 2 gamma. This is a significant improvement compared to the existing bounds gamma + 1. Several exponent matrices of protograph-based LDPC codes with chordless short cycles are proposed for any type of protographs, single-edge and multi-edge. These numerical results as well as simulations show that the removal of short cycles with a chord improves previous results in the literature.