Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

86 Chapitre 2 : Hybrid LDPC Codes

Therefore

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

2.7.3 LM-invariance

Card(A ∈ E k,l : Ai = j)

Card(E k,l )

= 1

q l − 1

Lemma 7. If a probability-vector random variable Y of size q 2 is LM-invariant, then

for all (i, j) ∈ [0, q 2 − 1] × [0, q 2 − 1], the random variables Y i and Y j are identically

distributed.

Proof: For any (q 1 , q 2 ), q 1 < q 2 , T 1,2 denotes the set of all truncations from G(q 2 ) to

G(q 1 ). We assume Y LM-invariant. A −1 and B −1 denote two truncations independently

arbitrary chosen in T 1,2 . For any l and k in [0, q 2 − 1], we can choose extension A such

that l ∈ Im(A) and A −1 l is denoted by i. Also, we choose B such that Bi = k. Y

LM-invariant implies

∀(i, A −1 , B −1 ) ∈ [0, q 1 − 1] × T 1,2 × T 1,2 , P(Y ×A−1

i

= x) = P(Y ×B−1

i = x)

This is equivalent to

P(Y Ai = x) = P(Y Bi = x)

and hence

P(Y l = x) = P(Y k = x), ∀(l, k) ∈ [0, q 2 − 1] × [0, q 2 − 1]

Lemma 8. A probability-vector random variable Y of size q 2 is LM-invariant if and

only if there exist q 1 and a probability-vector random variable X of size q 1 such that

Y = ˜X.

Proof: Let us first assume Y = ˜X and prove that Y is LM-invariant. This means that

we want to prove that for any (B, C) ∈ E 1,2 × E 1,2 , Y ×B−1 and Y ×C−1 are identically

distributed.

By hypothesis Y = X ×A , with A uniformly chosen in E 1,2 . We define the matrix α A of

size q 2 × q 1 . This matrix is such that Y = α A X and is defined by

∀j = 0 . . .q 1 − 1, ∀i = 0 . . .q 2 − 1, α A (i, j) = 1 if i=Aj

Thus, vector Y truncated by any linear map B is expressed by:

The same holds for linear map C:

Y ×B−1 = α T B α AX

Y ×C−1 = α T Cα A X

= 0 otherwise

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