Cryptography Reference
In-Depth Information
4.2.2 Implementing the encoder
The schematic diagram of an encoder for Reed-Solomon codes is quite similar
to that of an encoder for cyclic codes with binary symbols, but the encoder
must now carry out multiplications between
q
-ary symbols and memorize
q
-ary
symbols.
As an example, we have shown in Figure 4.4 the schematic diagram of the
encoder for the Reed-Solomon code treated in the example above.
Figure 4.4 - Schematic diagram of the encoder for the RS code (15,11).
4.3
Decoding and performance of codes with bi-
nary symbols
4.3.1 Error detection
Considering a binary symmetric transmission channel, the decoder receives bi-
nary symbols assumed to be perfectly synchronized with the encoder. This
means that the splitting into words having
n
symbols at the input of the de-
coder corresponds to the splitting used by the encoder. Thus, in the absence of
errors, the decoder sees codewords at its input.
Let us assume that codeword
c
is transmitted by the encoder and let
r
be
the word of
n
symbols received at the input of the decoder. Word
r
can always
bewrittenintheform:
r
=
c
+
e
where
e
is a word whose non-null symbols represent the errors. A non-null
symbol of
e
indicates the presence of an error in the corresponding position of
c
.
Errors are detected by using the orthogonality property of the parity check
matrix with the codewords and calculating the quantity
s
called the error syn-
drome.
s
=
rH
T
=(
c
+
e
)
H
T
=
eH
T
