Cryptography Reference
In-Depth Information
120
100
80
60
40
20
50
100
150
200
250
Figure 9.4 - Parity check matrix of a regular (3,6) LDPC code of size
n
= 256
and
rate
R
=0
,
5
.
and the total number
E
of 1s in matrix
H
. For example,
λ
(
x
)=0
,
2
x
4
+0
,
8
x
3
indicates a code where
20%
of the 1s are associated with variables of degree 5
and
80%
with variables of degree 4. Note that, by definition,
λ
(1) =
λ
j
=1
.
Moreover, the proportion of variables of degree
j
in the matrix is given by
λ
j
/j
λ
j
=
k
λ
k
/k
Symmetrically, the irregularity profile of the parities is represented by the poly-
nomial
ρ
(
x
)=
ρ
p
x
p−
1
,coecient
ρ
p
being equal to the ratio between the
accumulated number of 1s in the rows (or parity) of degree
p
and the total num-
ber of 1s denoted
E
. Similarly, we obtain
ρ
(1) =
ρ
j
=1
. The proportion
ρ
p
of columns of degree
p
in matrix
H
is given by
ρ
p
/p
k
ρ
p
=
ρ
k
/k
Irregular codes have more degrees of freedom than regular codes and it is
thus possible to optimize them more eciently: their asymptotic performance
is better than that of regular codes.
Coding rate
Consider a parity equation of degree
d
c
.
It is possible to arbitrarily fix the
values of the
d
c
−
1
first bits; only the last bit is constrained and corresponds
to the redundancy. Thus, in a parity matrix
H
of size (
m
,
n
), each of the
m
rows corresponds to 1 redundancy bit. If the
m
rows of
H
are independent, the
code then has
m
redundancy bits. The total number of bits of the code being