Cryptography Reference
In-Depth Information
Figure 4.6 - Performance of the soft input decoding of the (15,7) BCH code. 4-PSK
transmission on a Gaussian channel.
4.4
Decoding and performance of codes with non-
binary symbols
4.4.1 Hard input decoding of Reed-Solomon codes
Hard input decoding algorithms make it possible to decode Reed-Solomon (RS)
codes and BCH codes with binary symbols. We begin by presenting the principle
of decoding RS codes then we treat the case of BCH codes using binary symbols
as a particular case of decoding RS codes.
Assuming that
c
(
x
)
is the transmitted codeword, then for a channel with
discrete input and output, the received word can always be written in the form:
r
(
x
)=
c
(
x
)+
e
(
x
)
with:
+
e
j
x
j
+
+
e
n−
1
x
n−
1
,
e
(
x
)=
e
0
+
e
1
x
+
···
···
j
∈
F
q
∀
j
When
e
j
=0
there is an error in position
j
.
It was seen above that the generator polynomial of an RS code or of a BCH
code (with
l
=1)
correcting
t
errors had the roots
α,
,α
j
,
,α
2
t
and that the
···
···
