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When decoding a cyclic code, an alternative to computing the syndrome by dividing the received -tuple by the generator polynomial is to compute the product , mod , with the check polynomial . It is shown in this paper that the form of the product can be predicted in terms of general code parameters and corresponds closely to the burst error from which it is derived. By using the properties of the product sequence, a burst-error decoder is derived in such a way that a family of potentially fast burst-error decoders can be constructed. Another important application of the proposed technique concerns decoder implementation for the correction of a synchronization error (slip) when the coset code technique is used. It is shown that slip correction can be implemented so that both the magnitude and direction of slip are determined by examining only one received -tuple.