부산대학교 도서관

There is a concept in graph theory known as a chord which has not been considered before in relation to trapping sets of Tanner graphs. A chord of a cycle is an edge outside the cycle which connects two vertices of that cycle. It is proved that short cycles with a chord are the root of several trapping sets and eliminating them increases the minimum distance $d_{\min }$ of a code. We provide new analytic lower bounds on $d_{\min }$ of LDPC codes with girths 6 and 8 and column weight $\gamma $ in which the short cycles are all chordless. We prove, analytically, that $d_{\min }\geq 2\gamma $ for girth 6 and $d_{\min }\geq \frac {3(\gamma -1)^{2}}{\gamma \ln \gamma -\gamma +1}$ for girth 8. Comparing these bounds with the existing bound $\gamma +1$ for girth-6 LDPC codes shows the positive and significant influence of eliminating these cycles. A method to construct protograph-based LDPC codes with different girths and free of short cycles with a chord is given which is applicable to any type of protographs, simple and multi-edge, regular and irregular. The conditions to remove small trapping sets from the Tanner graph of a multi-edge QC-LDPC code are given. Numerical results indicate that the application of our method to QC-LDPC codes improves existing results.