Cyclic codes and quadratic residue codes over Z4

A set of n-tuples over Z₄ is called a code over Z₄ or a Z₄ code if it is a Z₄ module. We prove that any Z₄-cyclic code C has generators of the form (fh, 2fg) where fgh=xⁿ-1 over Z₄ and |C|=4^{deg g} 2^{deg h}. We also show that C^⊥ has generators of the form (g^*h^*, 2f^*g^*). We show that idempotent generators exist for certain cyclic codes. A particularly 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. We show that the nonlinear binary images of the extended QR codes of lengths 32 and 48 have higher minimum weights than comparable known linear codes