2017/09/04 by Farzane Amirzade, Amirzade, Farzane, Mohammad‐Reza Sadeghi +1
Computer Science · Engineering · #Advanced Wireless Communication Techniques #Cooperative Communication and Network Coding #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.1709.00825
openalex publication_date 2017/09/04 · openalex created_date 2017/09/15 · openalex updated_date 2026/07/28
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≤ n≤ 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.