Cyclic Codes For Correcting Errors
Now, we will begin the discussion of cyclic codes explicitly with error detection and correction. Cyclic codes can be used to correct errors, like Hamming codes as a cyclic codes can be used for correcting single error. Likewise, they are also used to correct double errors and burst errors. All types of error corrections are covered briefly in the further subsections.
The (7,4) Hamming code has a generator polynomial . This polynomial has a zero in Galois extension field at the primitive element, and all codewords satisfy . Cyclic codes can also be used to correct double errors over the field . Blocklength will be equal to and primitive elements and as zeros in the because we are considering the case of two errors here, so each will represent one error.
The received word is a polynomial of degree given as
where can have at most two nonzero coefficients corresponding to 2 errors.
We define the Syndrome Polynomial, as the remainder of polynomial when divided by the generator polynomial i.e.
= as is zero.
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