General description Encoding and decoding An linear code is called a cyclic code if a cyclic shift of any codeword is also a codeword. The shift may be leftwards or rightwards, by any number of bits.
The cyclic code nature can be utilized when encoding and decoding, using linear-feedback shift registers. Hardware implementations are not discussed.
Consider the Hamming code (7,4), defined by parity-check sums (6.4.7). Its codewords are presented in Table 6.4.2. Then, consider the codeword . Its cyclic shift, rightwards by one bit: is not the codeword, as its syndrome . Therefore, such Hamming code is not a cyclic code. Consider an other code, obtained from the Hamming code by permutation of columns, such that systematic code is obtained. The code is defined by the following parity-check matrix and parity-check equations:
All code words are presented in Table 7.1.1. As can be observed, the obtained code is the cyclic one, there are four cycles in this code. Assignment of codewords to cycles is outlined in the last column of this Table. All-0 word gives the 1 st cycle, all-1 word gives the 2 nd cycle, cycles III and IV are shown in Fig.7.1.1.
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