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2. Annex R(x) to U(JC). Then we have the codeword. The preceding operation can be performed by the circuit of Fig. 5, which is a revised version of the division circuit of Fig. 2. 9ul9u0 into the circuit. , uv u0 at the output terminal. When the last mes­ sage symbol u0 has been fed in, the remainder R(x) is left in the delay elements. 3. Decoding Cyclic Codes Since a cyclic code is a linear code, it is decoded by making use of the syndrome, as described in Sec. 2 of Chapter 2. However, the syndrome polynomial to be defined shortly is usually used for decoding cyclic codes instead of the syndrome vector defined earlier.

1 is easily realized by making use of the syndrome if the code is linear. Takahiro Yamada 34 If a set of error vectors that must be detected is given, we have to construct a parity check matrix H so that HeT Φ 0 for any error vector e in the set. 16 Let's consider the (3,2) code over GF(2) with parity check matrix H= ( 1 1 1 ) . 57) T The codewords of this code are all vectors x such that Hx = 0, that is, C = {000,101,011,110}. 5. As described there, this is a single-error-detecting code. Error detection is performed by check­ ing if the syndrome is 0.

Peterson, W. , and Weldon, E. , Jr. (1972). "Error-Correcting Codes," 2nd ed. MIT Press, Cambridge, Massachusetts. 1. 1. 2. 3. 2. 1. 2. 3. 3. 4. 1. 2. 3. 5. 1. 2. 3. 1. Cyclic Codes Cyclic codes, an important class of codes, are used in many systems because they are easy to implement. In the first section of this chapter, we define cyclic codes and study their encoders and decoders. 1. Definition of Cyclic Codes Messages and codewords have been expressed as vectors so far in this book. However, we express them as polynomials for cyclic codes.

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