Cryptography Reference
In-Depth Information
n
k
t
g
(
x
)
7
4
1
13
15
11
1
23
31
26
1
45
63
57
1
103
127
120
1
211
255
247
1
435
511
502
1
1021
1023
1013
1
2011
2047
2036
1
4005
4095
4083
1
10123
Table 4.2 - Parameters of some Hamming codes.
1. Parity check code
Let us consider a BCH code with
l
=0
and
t
=0
.Itsgenerator
polynomial,
g
(
x
)=(
x
+1)
has only one root
α
0
=1
.Thiscodeuses
only one redundancy symbol and the
c
(
x
)
words of this code satisfy
the condition:
c
(
α
0
)=
c
(1) = 0
This code, which is cyclic since
(
x
+1)
divides
(
x
n
+1)
,isaparity
check code with parameters
n
=
k
+1
,k,t
=0
. Thus, every time we
build a BCH code by selecting
l
=0
, we introduce into the genera-
tor polynomial a term in
(
x
+1)
and the codewords are of even weight.
2. Cyclic Redundancy Code (CRC)
Another example of a BCH code for which
l
=0
,istheCRCused
for detecting errors. A CRC has a constructed distance of 4 (
t
=1
)
and its generator polynomial, from above, is therefore equal to:
g
(
x
)=(
x
+1)
m
α
(
x
)
α
being a primitive element,
m
α
(
x
)
is a primitive polynomial and thus
the generator polynomial of a CRC is a code equal to the product of
(
x
+1)
by the generator polynomial of a Hamming code.
g
CRC
(
x
)=(
x
+1)
g
Hamming
(
x
)
The parameters of a CRC are therefore:
n
=2
m
k
)=
m
+1;
k
=2
m
−
1; (
n
−
−
m
−
2