I am a bit confused about this example in Huffman and Pless's Fundamentals in Error-Correcting Codes. This can be found in Chapter 12:
Let $\mathcal{C}$ be a nonempty subset of $\mathbb{Z}{}^4_4$ with 16 elements/codewords as follows: \begin{equation*} \begin{array}{cccccccc} 0000 & 1113 & 2222 & 3331 & 0202 & 1311 & 2020 & 3133 \\ 0022 & 1131 & 2200 & 3313 & 0220 & 1333 & 2002 & 3111 \end{array} \end{equation*}
Show that every codeword of C can be written uniquely in the form \begin{equation*} x\mathbf{c}_1 + y\mathbf{c}_2 + z\mathbf{c}_3, \end{equation*} where \begin{equation*} \mathbf{c}_1 = 1113, \quad \mathbf{c}_2 = 0202, \quad \mathbf{c}_3 = 0022, \end{equation*} $x \in \mathbb{Z}_4$ and $y,z \in \mathbb{Z}_2$.
Use (1) to show that $\mathcal{C}$ is a $\mathbb{Z}_4$-linear code.
I find the problem baffling since it is (somehow?) not possible to operate (addition or multiplication) an element of $\mathbb{Z}_4$ with an element of $\mathbb{Z}_2$. Can anybody help me understand the problem? Thanks and more power!