B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

856 ERROR CORRECTING CODES
is known as the column weight Yj · In LDPC codes, both row and column weights are much
smaller than the code length n, that is,
p; « n Yj « n
For regular LDPC codes, all rows have equal weight Pi = p and all columns have equal
weight y; = y. For irregular LDPC codes, the row weights and column weights do vary and
typically exhibit certain weight distributions. Regular codes are easier to generate, whereas
irregular codes with large code length may have better performance.
Bipartite (Tanner) Graph
A Tanner graph is a graphic representation that can conveniently describe a linear block code;
This bipartite graph with incidence matrix H was introduced by R. M Tanner in 1981. 18
Consider an (n, k) linear block code. There are n coded bits and n - k parity bits. The Tanner
graph of this linear block code has 11 variable nodes corresponding to the n code bits. These
11 variable nodes are connected to their respective parity nodes (or check nodes) according to
the ls in the parity check matrix H. A variable node (a column) and a check node (a row) are
connected if the corresponding element in H is a 1. Because H is sparse, there are only a few
connections to each variable node or check node. These connections are known as edges. Each
row represents the connection of a check node, and each column represents the connection of
a variable node. For LDPC codes, if the ith row of H has row weight of Pi, then the check
node has p; edges. If column j has column weight of y;, then the variable node has y; edges.
We use an example to illustrate the relationship between H and the Tanner graph.
Example 14. 9 Consider a Hamming (7, 4, 3) code with parity check matrix
1
1 !]
1 0 1 0 0 1
(14.71)
Determine its Tanner graph.
This code has 7 variable nodes and 3 check nodes. Based on the entries in H, each check
node is connected to 4 variable nodes. The first row of H corresponding the connection to
check node 1. The nonzero entries of H mark the connected variable nodes. The resulting
Tanner graph is shown in Fig. 14.23.
Fi g ure 14.23
Tanner graph of
the (7, 4, 3)
Hamming code.
Variable nodes
Check nodes