Cyclic codes

Absract

In this chapter R⁽ⁿ⁾ will denote the n-dimensional vector space over GF(q). We shall make the restriction (n,q) = 1. Consider the ring R of all polynomials with coefficients in GF(q), i.e. (GF(q)[x],+, ). Let S be the principal ideal in R generated by the polynomial xⁿ-1, i.e. S := (({xⁿ-1}),+, ). R/S is the residue class ring R mod S, i.e. (GF(q)[x] mod ({xⁿ-1},+, ). The elements of this ring can be represented by polynomials of degree < n with coefficients in GF(q). The additive group of R/S is isomorphic to R⁽ⁿ⁾. An isomorphism is given by associating the vector a = (a_O,a₁, ... ,a_n-1) with the polynomial a_O + a₁x + ... + a_n-1x^n-1.