Reed-Solomon Error Correcting Codes
1 Galois Fields
The field GF(2 m ) is constructed with a primitive polynomial of order m.
2 Encoding
The basic Reed-Solomon (n, k)-code will correct at least t errors, where n = 2 m − 1
and n − k = 2t. Our example will be a (7,3)-code over GF(2 3 ), which will correct
up to 2 errors.
3 Decoding
3.1 Error Values and Error Locators
3.2 Options for Decoding ( Using the Berlekamp-Massey Algorithm )
3.3 Calculating Syndromes
Calculate the syndromes
S i = r(α i ) for i = 1, 2, . . . , 2t.
Then the syndrome polynomial is
S(x) = S 2t x 2t − 1 + · · · + S 1 .
3.4 Error Locator Polynomial
3.5 The Key Equation
3.6 The Euclidean Algorithm
3.7 The Berlekamp-Massey Algorithm
3.8 Chien Search
3.9 Forney’s Algorithm
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