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AN2407

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

Freescale Semiconductor 

The redundant symbols are obtained from the redundant polynomial p(x), which is the remainder

obtained by dividing x2Tm(x) by the generator polynomial g(x):

p(x) = (x2Tm(x))modg(x)

Theory

where is the generator polynomial. We choose the most frequently used generating polynomial:

g(x) = (x + αp1)(x + αp2)(x + αp3)…(x + αp2T)

g(x) = (x + α)(x + α2)(x + α3)…(x + α2T)

The code-word polynomial c(x) is defined as follows:

c(x) = x2Tm(x) + p(x)

Since in GF(2m) algebra, plus (+), and minus (–) are the same, the code word actually equals the polynomial

x2Tm(x) minus its remainder under division by g(x). It follows that c(x) is a multiple of g(x). Since there is a total of

2mK different possible messages, there are 2mK different valid code words at the encoder output. This set of 2mK

code words of length N is called an (N,K) block code.

2.2.2 Decoding

When a received block is input to the decoder for processing, the decoder first verifies whether this block appears

in the dictionary of valid code words. If it does not, errors must have occurred during transmission. This part of the

decoder processing is called error detection. The parameters necessary to reconstruct the original encoded block

are available to the decoder. If errors are detected, the decoder attempts a reconstruction. This is called error

correction. Conventionally, decoding is performed by the Petersen-Gorenstein-Zierler (PGZ) algorithm, which

consists of four parts:

1. Syndromes calculation.

2. Derivation of the error-location polynomial.

3. Roots search.

4. Derivation of error values.

The error-location polynomial in this implementation is found using the Berlekamp-Massey algorithm, and the

error values are obtained by the Forney algorithm. The four decoding parts are briefly outlined as follows:

1. Syndromes calculation: From the received block, the received polynomial is reconstructed, denoted as

c(x). The received polynomial is the superposition of the correct code word c(x) and an error polyno-

mial e(x):

r(x) = c(x) + e(x)

The error polynomial is given in its most general form by:

e(x)

=

ei0xi0

+

ei1xi1

+

e i2 x i2 …,

ei(ν

i

x

(

ν

– 1)

–

1)

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

Freescale Semiconductor

7

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