A transmitted code vector may be perturbed by noise, then the vector received may be a corrupted version of the transmitted code vector. A codeword with polynomial representation U(X) is transmitted and Z(X) is the corrupted version of V(X), then it must be a multiple of the generator polynomial g(X), that is, U(X) = m(X) g(X) and Z(X) can be written Z(X)= U(X) + e(X), where e(X) is the error pattern polynomial. This is accomplished by calculating the syndrome of received polynomial.

The syndrome S(X) is equal to the remainder resulting from dividing Z(X) by g(X), that is, Z(X) = q(X)g(X) + S(X). And the other hand, S(X) is exactly the same polynomial obtained as remainder of e(X) modulo g(X). Thus the syndrome of the received polynomial Z(X) contains the information needed for correction of error pattern.

When the syndrome is an all-zeros vector, the received vector to be a valid code vector. When the syndrome is a nonzero vector, the received vector is pertubed code vector and errors have been detected. The procedure for error detection is as follows. The received vector is first stored in a buffer. It is subjected to devide by g(X) operation, the division can be carried out very efficiently by a shift register circuit. The remainder in the shift register is then compared with all the possible syndromes. This set of syndromes corresponds to the set of correctable error patterns. If a syndromes match is found, the error is subtracted out from the received vector. The correct version of the received vector is then pass on the next stage of the received unit for further processing.

Key words: error detection, transmitted, received code vector, Syndrome, pertubed

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