Cyclic codes over GF(q) with simple orbit structure (Q1821757)

Let C be a cyclic [n,k] code over \(F=GF(q)\), with generator g(x) and parity check polynomial h(x). C is said to have simple orbit structure if any polynomial \(g_ 1(x)\) that generates the ideal C in \(F[x]/(x^ n- 1)\) must have the form \(g_ 1(x)=ax^ ig(x)\) with \(a\in F\). It is shown how to compute the weight distribution of C in this case. The main result is the characterization of the polynomials h(x) corresponding to codes with simple orbit structure. In the binary case h(x) is the product of primitive polynomials of relatively prime degrees, possibly multiplied by \((X+1)^ 2\) or \((X+1)^ 3\) if one allows even word length.