  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.
