參數(shù)資料
型號： AN2407
廠商： 飛思卡爾半導(dǎo)體(中國)有限公司
英文描述： Reed Solomon Encoder/Decoder on the StarCore SC140/SC1400 Cores, With Extended Examples
中文描述： 里德所羅門編碼器/的StarCore SC140/SC1400核心解碼器，以擴(kuò)展實例
文件頁數(shù)： 4/48頁
文件大小： 306K
代理商： AN2407
Reed Solomon Encoder/Decoder on the StarCore SC140/SC1400 Cores, With Extended Examples, Rev. 1
4
Freescale Semiconductor
Theory
2.1.1 Binary Field, GF(2)
The simplest Galois field is GF(2). Its elements are the set {0,1} under modulo-2 algebra. Addition and subtraction
in this algebra are both equivalent to the logical XOR operation. The addition and multiplication tables of GF(2)
are shown in
Figure 2
.
Figure 2.
Addition (XOR) and Multiplication Tables of GF(2)
There is a one-to-one correspondence between any binary number and a polynomial in that every binary number
can be represented as a polynomial over
GF(2)
, and
vice versa
. A polynomial of degree
D
over
GF(2)
has the
following general form:
where the coefficients
f
0
,...,
f
D
are taken from
GF(2)
. A binary number of
(N+
1
)
bits
can be represented as an
abstract polynomial of degree
N
by
taking the coefficients equal to the bits and the exponents of
x
equal to the bit
locations.
For example, the binary number 100011101 is equivalent to the following polynomial:
x
2
x
3
+
+
The bit at the zero position (the coefficient of
x
0
) is equal to 1, the bit at the first position (the coefficient of
x
) is
equal to 0, the bit at the second position (the coefficient of
x
2
) is equal to 1, and so on. Operations on polynomials,
such as addition, subtraction, multiplication and division, are performed in an analogous way to the real number
field. The sole difference is that the operations on the coefficients are performed under modulo-2 algebra. For
example, the multiplication of two polynomials is as follows:
This result differs from the result obtained over the real number field (the middle expression) due to the XOR
operation (the + operation). The terms that appear an even number of times cancel out, so the coefficients of x
5
and
x
7
are not present in the end result.
2.1.2 Extended Galois Fields GF(2
m
)
A polynomial
p(x)
over GF(2) is defined as
irreducible
if it cannot be factored into non-zero polynomials over
GF(2) of smaller degrees. It is further defined as
primitive
if
n = (x
n
+ 1)
divided by
p(x)
and the smallest positive
integer
n
equals 2
m
–1, where
m
is the polynomial degree. An element of
GF(2
m
)
is defined as the root of a
primitive polynomial
p(x)
of degree
m
. An element
α
is defined as primitive if
Addition
+
0
1
0
0
1
1
1
0
Multiplication
x
0
1
0
0
0
1
0
1
f x
( )
f
0
f
1
x
f
2
x
2
f
3
x
3
…
f
+
D
x
D
+
+
+
=
100011101
1
x
4
x
8
+
+
1
x
2
x
3
x
4
+
+
+
(
)
x
3
x
5
+
(
)
x
3
x
5
x
5
x
6
x
7
x
7
x
8
x
9
x
3
x
6
x
8
x
9
+
+
+
=
+
+
+
+
+
+
+
=
α
imod(2
)
m–1
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