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AN2407

Reed Solomon Encoder/Decoder on the StarCore™ SC140/SC1400 Cores, With Extended Examples

Freescale Semiconductor 

Theory

Table 1. Exponential-to-Binary Table for ADSL Systems

p

αp

5

0x20

6

0x40

7

0x80

8

0x1D

9

0x3A

10

0x74

...

...

253

0x47

254

0x8E

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(2m). A block consists of information symbols and added redundant symbols. The total number of symbols is

the fixed number 2m –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 nerrors and the number of errors with known locations as nerasures, the Reed-Solomon algorithm guarantees to

correct a block, provided that the following is true: 2nerrors + nerasures ≤ 2T, 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 N = 2m – 1 symbols

each. The encoder divides the information sequence into message blocks of K ≡ N – 2T symbols. Each message

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 2T 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.

K Message Symbols

2T Redundant Symbols

N = K+2T Block Symbols

Figure 3. Block Structure

Reed Solomon Encoder/Decoder on the StarCore™ SC140/SC1400 Cores, With Extended Examples, Rev. 1

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