Cyclic codes and quadratic residue codes over Z4

We prove that any Z₄-cyclic code has generators of the form (fh, 2fg) where fgh=xⁿ-1 over Z₄. From this we can easily find the order of the code and generators of the dual. A particular interesting family of Z₄-cyclic codes are quadratic residue codes. We define such codes in terms of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of quadratic residue codes over a field