From Skew-Cyclic Codes to Asymmetric Quantum Codes

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.