Generator polynomial Consider nonzero codeword of a cyclic code having the smallest degree of its polynomial, i.e. having the longest all-zero sequence at first (leftmost) bits: . For the considered cyclic code (7,4), defined by parity-check matrix (7.1.1), : . Each cyclic shift of this word is also a codeword. That way, next code words can be obtained from (7.1.7), index is an integer from 1 to .
. For the considered cyclic code (7,4), these words are: . A linear code satisfies condition (5.2.1), mod-2 sum of any two codewords is a codeword. Therefore, words defined by (7.1.7), added mod-2 to , give next words:
For the considered cyclic code (7,4), these words are: .
Any other mod-2 sum of codewords defined by (7.1.6), (7.1.7), (7.1.8) is a codeword. That corresponds to multiplication of by any other polynomial of the degree less than or equal . For the considered cyclic code (7,4), the remaining eight nonzero codewords are obtained.
Products of an cyclic code nonzero codeword described by a polynomial of the smallest degree , with every polynomial of the degree less than or equal to , produce the complete set of nonzero codewords. Such polynomial is called the generator polynomial . For the considered cyclic code . In an cyclic code, product of with every polynomial of degree or less is a codeword. Then, the following conclusion can be drawn. Any nonzero code word (word index is omitted for simplicity of description) of a cyclic code can be divided by the generator polynomial , with the reminder equal zero. In another words, any codeword polynomial has the generator polynomial as a factor:
. However, not every polynomial can be used to generate an cyclic code.
A polynomial generates an cyclic code, if divides the following polynomial: , In another words, a cyclic code generator polynomial is a factor of polynomial (7.1.11). For large n , polynomial (7.1.11) may have many factors of degree , some number of cyclic codes may be produced. Some of them are “good” codes, some are “bad” codes. Selection of a “good” code is a difficult problem, several classes have been discovered.
For the considered cyclic code of the block-length ,
From division of the polynomial by the generator polynomial
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