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，以擴展實例
文件頁數(shù)： 6/48頁
文件大?。?/td> 306K
代理商： AN2407
Reed Solomon Encoder/Decoder on the StarCore SC140/SC1400 Cores, With Extended Examples, Rev. 1
6
Freescale Semiconductor
Theory
The zero element does not appear in the table since it deserves special attention (see
Section 4.3
,
Look-up Tables
).
Although multiplication is a complicated operation when performed bitwise, it is very simple if the exponential
representation is used. The converse is true for addition. Therefore, two types of look-up tables are useful: a log
table as shown in
Table 1
and an anti-log table that translates from binary to exponential representation.
2.2 Reed-Solomon Codes
Reed-Solomon codes are encoded and decoded within the general framework of algebraic coding theory. The main
principle of algebraic coding theory is to map bitstreams into abstract polynomials on which a series of
mathematical operations is performed. Reed-Solomon coding is, in essence, manipulations on polynomials over
GF(2
m
)
. A block consists of information symbols and added redundant symbols. The total number of symbols is
the fixed number
2
m
–1
. The two important code parameters are the symbol size
m
and the upper bound,
T
, on
correctable symbols within a block.
T
also determines the code rate, since the number of information symbols
within a block is the total number of symbols, minus
2T.
Denoting the number of errors with an unknown location
as
n
errors
and the number of errors with known locations as
n
erasures
, the Reed-Solomon algorithm
guarantees
to
correct a
block, provided that the following is true:
2n
errors
+ n
erasures
≤ 2
T,
where
T
is configurable
.
The current
implementation does not deal with erasures, and this document considers only error correction.
2.2.1 Encoding
When the encoder receives an information sequence, it creates encoded blocks consisting of
each. The encoder divides the information sequence into message blocks of
block is equivalent to a message polynomial
of degree
K –
1
,
denoted as
m(x).
In systematic encoding, the encoded
block is formed by simply appending
redundant symbols to the end of the
K
-symbols long-message block, as
shown in
Figure 3
. The redundant symbols are also called parity-check symbols.
symbols
symbols. Each message
Figure 3.
Block Structure
5
6
7
8
9
0x20
0x40
0x80
0x1D
0x3A
0x74
...
0x47
0x8E
10
...
253
254
Table 1.
Exponential-to-Binary Table for ADSL Systems
p
α
p
N
2
m
1
–
=
K
N
2
T
–
≡
2
T
K Message Symbols
2T Redundant Symbols
N = K+2T Block Symbols
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