Cryptography Reference
In-Depth Information
additions are performed and if there is no ambiguity in the notation, the term
modulo 2 may be omitted.
The encoder transforms the message containing four data bits: d =(
d
0
,
d
1
,
d
2
,
d
3
)
into a word of eight bits: c =(
d
0
,
d
1
,
d
2
,
d
3
,
r
0
,
r
1
,
r
2
,
r
3
)
, called
code-
word
. The codeword is therefore
separable
into one part that is the information
coming from the source, called the
systematic
1
part and a part added by the
encoder, called the
redundant
part. Any code producing codewords of this form
is called a systematic code. Most codes, in practice, are systematic but there is
one important exception in the family of convolutional codes (Chapter 5).
The law for the construction of the redundant part by the particular encoder
of Figure 1.1 can be simply written as:
3
r
j
=
d
j
+
d
p
(
j
=0
, ...,
3)
(1.3)
p
=0
Table 1.1 shows the sixteen possible values of c,thatis,theset{c}ofcodewords.
d
0
d
1
d
2
d
3
r
0
r
1
r
2
r
3
d
0
d
1
d
2
d
3
r
0
r
1
r
2
r
3
0000000010000111
0001111010011001
0010110110101010
0011001110110100
0100101111001100
0101010111010010
0110011011100001
0111100011111111
Table 1.1 - The sixteen codewords that the encoder in Figure 1.1 can produce.
We can first note that the coding law is
linear
: the sum of two codewords is
also a codeword. It is the linearity of relation (1.3) that guarantees the linearity
of the encoding. All the codes that we shall consider in what follows are linear as
they are all based on two linear operations: addition and permutation (including
shifting). Since the code is linear and the transmission of a codeword might be
affected by a process that is also linear (the addition of a perturbation: noise,
interference, etc.), the choice of a codeword, to explain or justify the properties
of the code, is completely indifferent. It is the "all zero" codeword that will
play this "representative" or reference role for all the codewords vis-à-vis the
general properties of the encoder/decoder pair. At reception, the presence of 1
will therefore be representative of transmission errors.
1
We can also say information part because it is made up of bits of information coming from
the source.
