Cyclic codes over GF(q) with simple orbit structure

AbstractLet C be a cyclic (n, k) code over F = GF(q) with generator polynomial g(x) and parity check polynomial h(x) = (xn − 1)/g(x). Let G denote the subgroup of the group of units of the algebra F[x]/(xn − 1) generated by x + (xn − 1) and the nonzero element sof F. This group acts on C by left multiplication and the elements in an orbit have the same weight. We say that C has simple orbit structure if the generators of C form a single G-orbit. This implies that the set of generators of each ideal in C is also a single G-orbit which makes it possible to determine the weight distribution of such a code with relative ease. The main result of this paper is the determination of all polynomials h(x) for which C has simple orbit structure. The proof proceeds by reducing the problem to a study of the group of units of the ‘parity check’ algebra F[x]/(h(x))