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Elements of Algebraic Coding Systems by Valdemar Cardoso da Rocha, Jr.

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Chapter 4

DECODING CYCLIC CODES

The decoding procedures for linear block codes are also applicable to cyclic codes. However, the algebraic properties associated with the cyclic structure allow important simplifications when implementing a decoder for a cyclic code, both for calculating the syndrome and for correcting errors. The syndrome computation consists in dividing by the generator polynomial g(x) the polynomial representing the word received from the channel. The remainder of this division is the syndrome, denoted by s(x). If s(x) = 0, the received word is accepted as being a codeword. Otherwise, i.e., if s(x) ≠ 0, we declare that errors have occurred. In this manner, it is clear that a circuit to detect errors with a cyclic code is rather ...

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