88 Chapitre 2 : **Hybrid** **LDPC** Codes

We finally obtain

This completes the proof.

P(X = x|A) = P(Y ×B−1 = x)

= P(Y ×A−1 = x)

= P(X = x)

□

Lemma 11 The product of two LM-invariant r**and**om probability-vectors is LM-invariant.

Proof: Let U **and** V be two LM-invariant r**and**om LDR-vectors of size q 2 . Let A **and**

B be any two linear maps from G(q 1 ) to G(q 2 ). Since U is LM-invariant, U ×A−1 **and**

U ×B−1 are identically distributed, by definition of LM-invariance. The same holds for

V. U ×A−1 V ×A−1 **and** U ×B−1 V ×B−1 are therefore identically distributed. Moreover, it

is clear that U ×A−1 V ×A−1 = UV ×A−1 , for any A. Hence, UV ×A−1 **and** UV ×B−1 is

LM-invariant. This completes the proof.

2.7.4 Proof of Theorem 3

X (k) denotes a probability-vector r**and**om variable of size q k . The j th component of the

r**and**om truncation of X (k) is denoted by

rt . The j th component of the r**and**om extension

of X (k) is denoted by

re

X (k)

j

X (k)

j

r**and**om truncation of X (k) is denoted by rt+re

X (k)

j

We define the operator D a by:

. The j th component of the r**and**om extension followed by a

D a (X (l) ) =

.

⎛

q

1

l −1

∑

⎜

√

E⎝

q l − 1

j=1

√ X(l) j

X (l)

0

The following equalities are hence deduced from the previous definitions:

⎛

⎞

⎛

re

X

E⎝√

(k)

j

⎠

re

= ∑ q k −1

⎞

1 ∑

Π(l|k) E⎝√ X(k) i ⎠

q

X (k)

l − 1

l

i=1 X (k)

0

0

⎛

⎞

⎛

⎞

rt

q E⎝√

⎠

1

l −1

∑

⎜

= E

√

⎝

X(l) j ⎟

⎠

q l − 1

⎛

E⎝√

X (l)

i

rt

X (l)

0

rt+re

X (k)

i

rt+re

X (k)

0

⎞

j=1

= D a (X (l) )

⎠ = ∑ l

1

Π(l|k)

q l − 1

X (l)

0

q k −1

∑

i=1

⎞

⎟

⎠

⎛

√

E⎝

√ X(k) i

X (k)

0

⎞

⎠

□