Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

2.7 Proofs of theorems in Chapter 2 85

Let call b = Aa.


P(W ×A−1 = w) =

P(W = x)

=

=

x:x 0 =w 0 ,x A1 =w 1 ,...,x A(q1 −1)=w q1 −1


x:x 0 =w 0 ,x A1 =w 1 ,...,x A(q1 −1)=w q1 −1


x:x 0 =w 0 ,x A1 =w 1 ,...,x A(q1 −1)=w q1 −1

= e −wa ∑

x:x 0 =w 0 ,x A1 =w 1 ,...,x A(q1 −1)=w q1 −1

= e −wa P(W ×A−1 = w +a )

e x b

P(W = x +b )

e −wa P(W = x +b )

P(W = x +b )

(2.31)

The last step is obtained by noting that:

∀i ∈ Im(A),

x +Aa

i

We have obtained equation (2.30).

= x Aa+i − x Aa = w a+A −1 i − w a = (w +b ) ×A−1

i

Please note that, in the sequel of this chapter, for all the proofs, we simplify the

notations as follows: For all group G(q), for all i ∈ [0, q − 1], the element α i is now

denoted by i. Also, since A is a linear map, the matrix of the application is also denoted

by A. Hence, for all linear map A from G(q 1 ) to G(q 2 ), A(α i ) = α j with α i ∈ G(q 1 ) and

α j ∈ G(q 2 ), is translated by Ai = j.

2.7.2 A useful lemma

Lemma 10 E k,l denotes the set of extensions from G(q k ) to G(q l ). For given k and l,


∀(i, j) ∈ [1, q k − 1] × [1, q l − 1],

Card(A ∈ E k,l : A −1 j = i)

Card(E k,l )

= 1

q l − 1

Proof: p k and p l denote log 2 (q k ) and log 2 (q l ), respectively.

Without any constraint to build a linear extension A from G(q k ) to G(q l ), except the one

of full-rank, we have 2 p l − 2 n−1 choices for the n th row, n = 1, . . .,p l .

For given i and j, with the constraint that Ai = j, we have 2 p l−b i

+ 2 ⌊ b i ⌋ 2 − 2 n−1 choices

for the n th row, n = 1, . . .,p l , where b i is the number of bits equal to 1 in the binary map

of α i . Thus, the number of A such that Ai = j is dependent only on i. Let say

Card(A ∈ E k,l : A −1 j = i) = β i

we have

q l −1


Card(A ∈ E k,l : Ai = j) = Card(E k,l )

j=1

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