Cryptography Reference
In-Depth Information
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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
 
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