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Advanced Coding for Fiber-Optics Communications Systems
Published in Andrew Ellis, Mariia Sorokina, Optical Communication Systems, 2019
An LDPC code can now be defined as an (n, k) linear block code whose parity-check matrix H has a low density of 1’s. A regular (wc, wr) LDPC code is a linear block code whose H matrix contains exactly wc 1’s in each column and exactly wr = wcn/(n–k) 1’s in each row, where wc << n–k. The code rate of the regular LDPC code is determined by R = k/n = 1−wc/wr. The graphical representation of LDPC codes, known as bipartite (Tanner) graph representation, is helpful in efficient description of LDPC decoding algorithms. A bipartite (Tanner) graph is a graph whose nodes may be separated into two classes (variable and check nodes), and where undirected edges may only connect two nodes not residing in the same class. The Tanner graph of a code is drawn according to the following rule: check (function) node c is connected to variable (bit) node v whenever element hcv in a parity-check matrix H is a 1. In an m × n parity-check matrix, there are m = n − k check nodes and n variable nodes. As an illustration, the Tanner graph of a regular (3,4) LDPC(36,13) code is shown in Fig. 6.3. The blue-circles represent variable (bit) nodes and green-squares represent the parity-check (function) nodes. In principle, the Tanner graph can be defined for any linear block code. For instance, the parity-check matrix for Hamming (15,11) code is given by
Error Control Coding
Published in Jerry D. Gibson, Mobile Communications Handbook, 2017
The significance of the Tanner graph is that it provides a “map” to the decoding algorithm that is typically used with LDPC codes—an algorithm called belief propagation or message passing. The details of the belief propagation algorithm can be found in any textbook on coding theory (e.g., [15,19]), but what follows is a brief overview.
LDPC Codes Based on Rational Functions
Published in IETE Journal of Research, 2021
Mohammad Gholami, Akram Nassaj
Low-density parity-check (LDPC) codes are the most promising class of linear block codes which for many data transmission and storage channels [1] perform very close to the Shannon capacity. By a -regular LDPC code, we mean the LDPC code whose parity-check matrix (PCM) has row and column weight L and J, respectively. To each PCM of an LDPC code, a Tanner graph [2], denoted by TG, is associated which is helpful to iteratively share the results of the local node decoding by passing them along the edges. The girth of an LDPC code is the length of the smallest cycle in its Tanner graph. The construction of LDPC codes with large girth is interesting, because of the accuracy of belief propagation, known as sum-product algorithm [3].