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
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