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From skew-cyclic codes to asymmetric quantum codes

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  • We introduce an additive but not $\mathbb F$4-linear map $S$ from $\mathbb F$4n to $\mathbb F$42n 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.
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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