Cryptography Reference
In-Depth Information
where erfc
(
x
)
denotes the complementary error function defined by erfc
(
x
)=
2
√
π
x
exp
−
t
2
d
t
.
d
min
is the minimum Hamming distance of the code associ-
∞
ated with the modulation considered, 2-PSK or 4-PSK in the present case.
N
(
d
)
represents the multiplicities of the code (see Section 1.5). In certain cases, these
multiplicities can be determined precisely (like for example simple convolutional
codes,Reed-Solomoncodes, BCH codes,etc. ...), and(3.21)caneasily beeval-
uated. For other codes, in particular turbo codes, it is not possible to determine
these multiplicities easily and we have to consider some realistic hypotheses in
order to get round the problem. The hypotheses that we adopt for turbo codes
and for LDPC codes are the following [3.1]:
Hypothesis 1: Uniformity
. There exists at least one codeword of weight
1
d
min
having an information bit
d
i
equal to "1", for any place
i
of the
systematic part (
1
•
≤
i
≤
k
).
•
Hypothesis 2: Unicity.
There is only one codeword of weight
d
min
such
that
d
i
=
"1".
•
Hypothesis 3: Non-overlapping
.The
k
codewords of weight
d
min
associated
with the
k
bits of information are distinct.
Using these hypotheses and limiting ourselves to the first term of the sum in
(3.21), the upper bound becomes an asymptotic approximation (low PERs):
2
erfc
d
min
R
E
b
k
PER
≈
(3.22)
N
0
The three hypotheses, taken separately, are more or less realistic. Hypotheses 1
and 3 are somewhat pessimistic as to the quantity of codewords at the minimum
distance. As for hypothesis 2, it is slightly optimistic. The three hypotheses to-
gether are suitable for an acceptable approximation of the multiplicity, especially
since imprecision about the value of this multiplicity does not affect the quality
of the final result. Indeed, the targeted minimum distance that we wish to deter-
mine from (3.22) appears in an exponential argument, whereas the multiplicity
is a multiplying coecient.
It is then possible to combine (3.22) with the results obtained in Section 3.3
which provide the signal to noise ratio limits. Giving
E
b
/N
0
the limit value
beyond which using a code is not worthwhile, we can extract from (3.22) the
MHD sucient to reach a PER at that limit value. Given, on the one hand,
that (3.22) assumes ideal (maximum likelihood) decoding and, on the other
hand, that the theoretical limit is not reached in practice, the targeted MHD
can be slightly lower than the result of this extraction.
Figure 3.6 presents some results obtained using this method.
1
The codes being linear, distance and weight have the same meaning.
