Some intuitive reasons given by others is that for each error, one parity symbol
is used to locate the error and the other parity symbol is used to find its correct value.
RS Encoder. Let??™s first discuss the RS encoder, the generating polynomial is represented as
(5.63)
(5.64)
Recall the degree of the generator polynomial is equal to the number of parity symbols. Alternatively,
the generator polynomials for a t error-correcting code must have as roots 2t consecutive powers of
. This is shown below.
(5.65)
Keeping along the lines of block codes, we will discuss the systematic form of the encoder. This is
accomplished by shifting the message polynomial, m(x), into the rightmost k stages of a code word
register and then appending a parity polynomial, p(x) in the leftmost n-k stages. The parity polynomial
is obtained by
(5.66)
xnk # m(x)
g(x) q(x)
p(x)
g(x)
g(x) q2t
j1
(x aj)
g(x) (x a) # (x a2)c(x a2t )
g(x) g0 g1x g2x2 c g2t1x2t1 g2tx 2t
t jdmin 1
2 k
t jn k
2 k
(n,k) (2m 1, 2m 1 2t)
PERFORMANCE IMPROVEMENT TECHNIQUES 255
where p(x) is defined as the remainder polynomial. Now we can write the code word, c(x), as
(5.67)
The encoder is similar to that presented for binary cyclic codes, except the mathematical operations
are now contained in GF(2m) dimension, instead of GF(2) for the binary codes.
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