Hybrid LDPC codes and iterative decoding methods - i3s

ensea.fr

Hybrid LDPC codes and iterative decoding methods - i3s

120Chapitre 4 : Two-Bit Message Passing Decoders for LDPC Codes Over the Binary Symmetric Channel

topological structure in the Tanner graph is generally difficult and is beyond scope of this

work.

Let us first give some additional definitions and notations. We define a path of length

d as a set of d connexe edges.

Definition 11 The neighborhood of order one of a node n is denoted by N 1 (n) and is

composed of all the nodes such that there exists an edge between these nodes and n. By

extension, N d (n) denotes the neighborhood of order d of node n, which is composed of

all the nodes such that there exists a path of length d between these nodes and n.

When T is a set of nodes, say T = ∪ i n i , then the order d neighborhood of T is N d (T) =

∪ i N d (n i ). Let v 1 1 , v1 2 and v1 3 the variable nodes on which the errors occur. Let V 1 =

{v 1 1 , v1 2 , v1 3 } and C1 = N 1 (V 1 ). For more easily readable notations, we denote N 2 (V 1 )\V 1

by V 2 and N 1 (V 2 )\C 1 by C 2 . Also we say that a variable is of type (p|q) when it has p

connections to V 1 and q connection to V 2 . The union of order d neighborhoods of all the

(p|q) type variable nodes is denoted by N d (p|q).

Now we state the main theorem.

Theorem 5 [Irregular expansion theorem] Let G be the Tanner graph of a columnweight-four

LDPC code with no 4-cycles, satisfying the following expansion conditions:

each variable subset of size 4 has at least 11 neighbors, each one of size 5 at least 13

neighbors and each one of size 8 at least 16 neighbors. Then the code can correct up

to three errors in the codeword, provided the two-bit decoder, with C = 2, S = 2 and

W = 1, is used.

For lighter notations, each expansion condition according to which each variable subset

of size i has at least j neighbors, will be denoted by “i → j expansion condition”.

Proof :

Remark: The proof can be followed more easily by looking at Tables 4.2 and 4.3. Table

4.2 draws the decision rule in terms of the numbers of messages −S, −W , W and S

going into a variable, when this variable node is decoded as 0 (resp. 1) and when the

channel observation is 1 (resp. 0). Table 4.3 draws update rule in terms of the numbers of

messages −S, −W , W and S going into the variable node v leading to different values

of the message w j (v, c) going out of v, when the received value is r v . We consider all the

subgraphs subtended by three erroneous variable nodes in a graph and prove that, in each

case, the errors are corrected. The possible subgraphs are shown in Figure 4.1. As shown,

five cases arise. In the reminder, we assume that the all-zero codeword has been sent.

Case 1: Consider the error configuration shown in Figure 4.1(a). In this case, variables

1, 2 and 3 send incorrect −W messages to their neighbors. They receive W messages

from all their neighboring check nodes, they are therefore decoded correctly. Error

occurs only if there exists a variable node with correct received value that receives four

−W messages from its neighboring check nodes (see Table 4.2). However, since variables

1, 2 and 3 are the only variables that send incorrect messages in the first iteration,

it is impossible to encounter such a variable node without introducing a 4-cycle. Hence,

More magazines by this user
Similar magazines