Lower bounds on the lifting degree of single-edge and multiple-edge QC-LDPC codes by difference matrices

In this paper, we define two matrices named as "difference matrices", denoted by $D$ and $DD$ which significantly contribute to achieve regular single-edge QC-LDPC codes with the shortest length and the certain girth as well as regular and irregular multiple-edge QC-LDPC codes. Making use of these matrices, we obtain necessary and sufficient conditions to have single-edge $(m,n)$-regular QC-LDPC codes with girth 6, 10 and 12. Additionally, for girth 6, we achieve all non-isomorphic codes with the minimum lifting degree, $N$, for $m=4$ and $5\leq n\leq 11$, and present an exponent matrix for each minimum distance. For girth 10, we provide a lower bound on the lifting degree which is tighter than the existing bound. More important, for an exponent matrix whose first row and first column are all-zero, we demonstrate that the non-existence of 8-cycles proves the non-existence of 6-cycles related to the first row of the exponent matrix too. All non-isomorphic QC-LDPC codes with girth 10 and $n=5,6$ whose numbers are more than those presented in the literature are provided. For $n=7,8$ we decrease the lifting degrees from 159 and 219 to 145 and 211, repectively. For girth 12, a lower bound on the lifting degree is achieved. For multiple-edge category, for the first time a lower bound on the lifting degree for both regular and irregular QC-LDPC codes with girth 6 is achieved. We also demonstrate that the achieved lower bounds on multiple-edge $(4,n)$-regular QC-LDPC codes with girth 6 are tight and the resultant codes have shorter length compared to their counterparts in single-edge codes. Additionally, difference matrices help to reduce the conditions of considering 6-cycles from seven states to five states. We obtain multiple-edge $(4,n)$-regular QC-LDPC codes with girth 8 and $n=4,6,8$ with the shortest length.