Cryptography Reference
In-Depth Information
the sphere of radius
√
nP
, the decoding pyramids form a partition of this same
sphere, and therefore the solid angle of this sphere
Ω
0
is the sum of all the solid
angles of the
Ω
i
pyramids. We can thus replace the solid angles
Ω
i
by the mean
solid angle
Ω
0
/
2
k
.
This progression, which leads to a lower bound on the error probability for an
optimal decoding of random codes on the Gaussian channel, is called the sphere-
packing bound because it involves restricting the coding to an
n
-dimensional
sphere and the effects of the noise to movements on this sphere.
Mathematical simplifications give an exploitable form of the lower bound on
the packet error rate (PER):
R
ln (
G
(
θ
i
,A
)sin
θ
i
)
2
A
2
AG
(
θ
i
,A
)cos
θ
i
k
1
ln (
PER
)
≥
−
−
arcsin
2
−R
θ
i
≈
(3.20)
≈
A
cos
θ
i
+
√
A
2
cos
2
θ
i
+4
/
2
A
=
2
RE
b
/N
0
G
(
θ
i
,A
)
These expressions link the size
k
of the messages, the signal to noise ratio
E
b
/N
0
and the coding rate
R
. For high values of
R
and for block sizes
k
lower than a
few tens of bits, the lower bound is very far from the real PER.
Asymptotically, for block sizes tending towards infinity, the bound obtained
by (3.20) tends towards the Shannon limit for a continuous input and output
channel such as presented in Section 3.2. In the same way as for the binary
input channel, if we wish to quantify the loss caused by the transmission of finite
length packets, we must normalize the values obtained by evaluating (3.20) by
removing the Shannon limit (3.15) from them, the penalty having to be null
when the packet sizes tend towards infinity. The losses due to the transmission
of finite length packets in comparison with the transmission of a continuous flow
of data are less in the case of a continuous input channel than in the case of a
binary input channel.
3.3.3 Some examples of limits
Figure 3.5 below gives an example of penalties caused by the transmission of
blocks of size
k
lower than 10000 bits, in the case of continuous input and in the
case of binary input. These penalty values should be combined with the values
of capacities presented in Figure 3.3, in order to obtain the absolute limits. As
we have already mentioned, this figure is to be considered with caution for small
values of
k
and high PER.
The results obtained concern the Gaussian channel. It is theoretically pos-
sible to consider the case of fading channels (Rayleigh, for example) but the
computations become complicated and the results very approximate.
