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 by which we achieve all non-isomorphic codes with the minimum lifting degree, N , for m = 4 and 5 ≤ n ≤ 11, and present an exponent matrix for each minimum distance. We attain the necessary and sufficient conditions to have a Tanner graph with girth 10. In this case we also 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. Moreover, necessary and sufficient conditions to have (m,n)-regular QC-LDPC codes with girth 12 as well as a lower bound on the lifting degree are 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 analytically prove that if ~ Bij is ij-th element of anm×n exponent matrix B then by taking three values A = max{2 ∑n j=1 ( | ~ Bij | 2 ) ; i = 1, 2 . . . ,m}, B = Manuscript received May ??, ????; revised November ??, ????. M.-R. Sadeghi is with the Department of Mathematics and Computer Science, Amirkabir University of Technology and F. Amirzade is with the Department of Mathematics, Shahrood University of Technology (e-mail: [email protected], [email protected]). Digital Object Identifier ????/TCOMM.????? September 5, 2017 DRAFT 2 max{2 ∑m i=1 ( | ~ Bij | 2 ) ; j = 1, 2 . . . , n} and C = max{ ∑n j=1 | ~ Bij | × | ~ Bi′j |; i 6= i ; i, i ∈ {1, 2 . . . ,m}} the minimum lifting degree is at least N = max{A,B,C}. 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. Additionaly 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. Index Terms Single-edge protographs, QC-LDPC codes, Girth, Difference matrices, Lifting degree.