94 Chapitre 2 : **Hybrid** **LDPC** Codes

• m is even: i · k is

even

odd

m/2

q

∑ ( m

2 2l)

m

l=0

∑

m/2−1

q

2 m l=0

times (2.46)

( m

2l+1)

times (2.47)

• m is odd: i · k is

even

odd

m−1

q

2∑ ( m

2 2l)

m

l=0

times (2.48)

m−1

q

2∑ ( m

2 2l+1)

times (2.49)

m

l=0

We complete the proof by showing that equations (2.46) **and** (2.47) are equal, so are (2.48)

**and** (2.49):

(1 − 1) m =

m∑

( ) m

=

k

k=0

⌊m/2⌋

∑

l=0

( m

−

2l)

⌊m/2−1⌋

∑

l=0

( ) m

= 0

2l + 1

Hence

⌊m/2⌋

∑

l=0

( m

=

2l)

⌊m/2−1⌋

∑

l=0

( ) m

2l + 1

□