On the classification of ${\mathbb{Z}}_4$-codes

In this note, we study the classification of ${\mathbb{Z}}_4$-codes. For some special cases $(k_1,k_2)$, by hand, we give a classification of ${\mathbb{Z}}_4$-codes of length $n$ and type $4^{k_1}{2}^{k_2}$ satisfying a certain condition. Our exhaustive computer search completes the classification of ${\mathbb{Z}}_4$-codes of lengths up to $7$.