RS Decoder. Next we move our attention to decoding RS codes. Assume the transmitted code word
is corrupted by errors such that the received signal is given as
(5.68)
where the error sequence is written as
(5.69)
(5.70)
Here the received code word has 2t unknowns, t of them are for the error locations and the other t are
for the error values themselves. Contrast this to the binary case where only the error locations are
unknown.
The transmitted code word is given as
(5.71)
It can be shown that the roots of g(x) are also the roots of c(x). Evaluating r(x) at each of the roots of
g(x) will produce zero when it is a valid code word. Hence the syndrome symbols, Si, are computed
as (for i 1, . . . , n k)
(5.72)
(5.73)
(5.74)
So far we have used the syndromes to determine if the received code word is valid or if an error is
present. If an error is present, we next need to determine its location. Let??™s assume there are p errors
in the code word, then the error polynomial is
(5.75)
We can define the error-locator polynomials as, (x)
(5.76)
where the roots of (x) are given as where are the error location numbers. Using
the same notation used in [1] and [25], the error-locator polynomial can also be written as
(5.
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