Algebraic Structure
Cyclic codes can be linked to ideals in certain rings. Let be a polynomial ring over the finite field . Identify the elements of the cyclic code C with polynomials in R such that maps to the polynomial : thus multiplication by x corresponds to a cyclic shift. Then C is an ideal in R, and hence principal, since R is a principal ideal ring. The ideal is generated by the unique monic element in C of minimum degree, the generator polynomial g. This must be a divisor of . It follows that every cyclic code is a polynomial code. If the generator polynomial g has degree d then the rank of the code C is .
The idempotent of C is a codeword e such that e2 = e (that is, e is an idempotent element of C) and e is an identity for the code, that is e c = c for every codeword c. If n and q are coprime such a word always exists and is unique; it is a generator of the code.
An irreducible code is a cyclic code in which the code, as an ideal, is maximal in R, so that its generator is an irreducible polynomial.
Read more about this topic: Cyclic Code
Famous quotes containing the words algebraic and/or structure:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“There is no such thing as a language, not if a language is anything like what many philosophers and linguists have supposed. There is therefore no such thing to be learned, mastered, or born with. We must give up the idea of a clearly defined shared structure which language-users acquire and then apply to cases.”
—Donald Davidson (b. 1917)