AN2407 查看數據表（PDF） - Freescale Semiconductor

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AN2407

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

Freescale Semiconductor 

Theory

where i0, i1 and so on denote the error location indices, and ν the actual number of errors that have

occurred. The 2T syndromes are obtained by evaluating the received polynomial r(x) at the 2T field

points:

α, α2, α3…, α·2· T

Since c(x) is a multiple of g(x), it has the following general form:

c(x) = q(x)g(x)

where q(x) is a message-dependent polynomial. It follows from the definition of g(x) that the following

field points:

α, α2, α3…, α2T

are roots of g(x). Hence c(x) vanishes at the 2T points and the syndromes:

S1, S2, S3…, S2T

contain only of the part consisting of the error polynomial e(x):

S1 = e(α)

S2 = e(α2)

S3 = e(α3)

…

S2T = e(α2T)

If all 2T syndromes vanish, e(x) is either identically zero, indicating that no errors have occurred

during the transmission, or an undetectable error pattern has occurred. If one or more syndromes are

non-zero, errors have been detected. The next steps of the decoder are to retrieve the error locations

and the error

error values

values from the syndromes. Denoting the actual number of errors as ν,αik

eik as Yk, the 2T syndromes S1 – S2T can then be expressed as follows:

as

Xk

and

the

S1 = Y1X1 + Y2X2 + Y3X3…YνXν

S2 = Y1(X1)2 + Y2(X2)2 + Y3(X3)2…Yν(Xν)2

S3 = Y1(X1)3 + Y2(X2)3 + Y3(X3)3…Yν(Xν)3

…

S2T = Y1(X1)2T + Y2(X2)2T + Y3(X3)2T…Yν(Xν)2T

Thus, there are 2T equations to solve that are linear in the error values Yk and non-linear in the error

locations Xk.

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

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Freescale Semiconductor

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