Lower Bounds on the Lifting Degree of QC-LDPC Codes by Difference Matrices

Abstract

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 B ij is ij-th element of an m×n exponent matrix B then by taking three values A = max{2 n j=1 | Bij | 2 ; i = 1, 2 . . . , m}, B =

Compact QC-LDPC Block and SC-LDPC Convolutional Codes for Low-Latency Communications

“…We have compared the obtained values of N and m h with those available in the literature. To the best of our knowledge, the design approaches that have produced till now the codes with minimum values of N and m h are those reported in [9], [11], [12], [15], [16].…”

“…Moreover, we relate the blocklength (for QC-LDPC block codes) and the constraint length (for SC-LDPC-CCs) of these codes to the latency and complexity of the decoding algorithms. Many theoretical lower bounds on the blocklength (constraint length) of QC-LDPC block codes (SC-LDPC-CCs) for several values of the girth have been proposed (see, for example, [8]- [11] for QC-LDPC block codes and [12] for SC-LDPC-CCs). Compared with numerical results, these bounds are tight when g = 6, 8, but provide a loose indication when g = 10, 12.…”

Two classes of QC-LDPC cycle codes approaching Gallager lower bound

“…That is, there exists only one isomorphic QC-LDPC cycle code with girth 12, i.e., the proposed codes [7]. Notice that Theorems 2 and 3 can be also proved based on the difference matrices in [9].…”

Protograph Based Low-Density Parity-Check Codes Design With Mixed Integer Linear Programming

“…In [35], a search loop is proposed that lists all labelings with desired girth among randomly generated choices, then numerical results of performance simulations are used as a code selection criterion. A concept of difference D and double difference DD matrices introduced in [36] facilitates computation of conditions for cycles elimination in cyclic liftings by reducing the number of inequalities that have to be tested. Using difference matrices, some numerical and analytical results have been provided, particularly concerning bounds on the lifting degrees, however these results involve mainly the fully-connected base graphs.…”