Cryptography Reference
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N 2
4
σ 2
therefore L (0)
j,p
σ 2 ,
We denote:
m (0)
j
2
σ 2 the average of the consistent Gaussian probability density of
variable c j of degree d v sent to parities e p of degree d c which are connected
to it,
=
the mean of messages Z ( n it )
j,p
μ ( n it )
p
.
To follow the evolution of the average m ( n it j during the iterations n it ,itthen
suces to take the mathematical expectation of Equations (9.22) and (9.23)
relative to the variable and parity processing , which gives:
Ψ μ ( n it )
p
m ( n it )
j
d c 1
with Ψ ( x )= E [tanh ( x/ 2)] ,x
N ( m, 2 m ) (9.27)
2
σ 2 +( d v
m ( n it +1)
j
1) μ ( n it )
p
=
(9.28)
Thus for a regular ( d v ,d c ) LDPC code and for a given noise with variance
σ 2 , Equations (9.27) and (9.28) enable us, by an iterative computation, to know
if the mean of the messages tends towards infinity or not. If such is the case,
it is possible to decode without errors with a codeword of infinite size and an
infinite number of iterations. In the case of an irregular code, it suces to make
the weighted mean on the different degrees of Equations (9.27) and (9.28).
The maximum value of σ for which the mean tends towards infinity, and
therefore for which the error probability tends towards 0, is the threshold of
the code. For example, the threshold of a regular code (3,6), obtained with the
density evolution algorithm, is σ max =0 . 8809 [9.13], which corresponds to a
minimum signal to noise ratio of E N 0 min =1 . 1 dB.
Another technique derived from extrinsic information transfer (EXIT)
charts 2 proposed by ten Brink [9.55, 9.56] enables the irregularity profiles to
be optimized. Whereas the density evolution algorithm is interested in the evo-
lution of the probability densities of the messages during the iterations, these
charts are interested in the transfer of mutual information between the input
and the output of the decoders of the constituent codes [9.56]. The principle of
these charts has also been used with parameters other than mutual information,
like the signal to noise ratio or error probability [9.3, 9.2].
It has also been
applied to other types of channels [9.15].
2 The principle of building EXIT charts is described in Section 7.6.3
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