There is exactly one Z2Z4-cyclic 1-perfect code

Let $\mathcal{C}$ be a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code of length $n

3$\mathrm{.}WeprovethatifthebinaryGrayimageof${\cal C}$,$C=\Phi({\cal C})$, is a 1-perfect nonlinear code, then $\mathcal{C}$ cannot be a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-cyclic code except for one case of length $n=15$. Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive 1-perfect code gives an extended 1-perfect code. We also prove that any such code cannot be ${\mathbb{Z}}_2{\mathbb{Z}}_4$-cyclic.