  We introduce an additive but not $\mathbb{F}_4$-linear map $S$ from
$\mathbb{F}_4^{n}$ to $\mathbb{F}_4^{2n}$ and exhibit some of its interesting
structural properties. If $C$ is a linear $[n,k,d]_4$-code, then $S(C)$ is an
additive $(2n,2^{2k},2d)_4$-code. If $C$ is an additive cyclic code then $S(C)$
is an additive quasi-cyclic code of index $2$. Moreover, if $C$ is a module
$\theta$-cyclic code, a recently introduced type of code which will be
explained below, then $S(C)$ is equivalent to an additive cyclic code if $n$ is
odd and to an additive quasi-cyclic code of index $2$ if $n$ is even. Given any
$(n,M,d)_4$-code $C$, the code $S(C)$ is self-orthogonal under the trace
Hermitian inner product. Since the mapping $S$ preserves nestedness, it can be
used as a tool in constructing additive asymmetric quantum codes.
