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In an (n,k) cyclic codę. productof g(x) withevery polynomial ofdegrce n-m-\ = k- \ or less is a codeword. Then. the following conclusion can be drawn.

Any non/.ero codę word u{.v) (word index / is omitted for simplicity of description) of a cyclic codę can be divided by the generator polynomial #(*), with the reminder equal zero. In another words, any codeword polynomial n{.v) has the generator polynomial as a factor:

w(ar)

£(.v)

r/U)©

rU)

*<*>

However. not every polynomial can be uscd to gcncrate an (n,k) cyclic codę.

A polynomial £(*) generatcs an (n,k) cyclic codę. if g(.v) dividcs the following polynomial:

jc" ©1,

In another words. a cyclic codę generator polynomial is a factor of polynomial (7.1.11). For large n, polynomial (7.1.11) may have many factors of degree m = n-k, some number of cyclic codes inay be produced. Sonie of them are “good” codes, sonie are “bad” codes. Selection of a “good” codę is a difficult problem, several classes have been discovered.

For the considered cyclic codę of the block-length n = 7,

a:⁷ © 1 <=> 10(XKXK)I