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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