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Erschienen in:
23.10.2022 | Original Research
Quantum codes from trace codes
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 2/2023
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Abstract
For an odd prime p and a positive integer m, let \({\mathbb {F}}_{p^{m}}\) be the finite field with \(p^{m}\) elements. For \(D_{1}=\{d\in {\mathbb {F}}_{p^{m}}^{*} : \textrm{Tr}_{e}^{m}(d^{2})=0\}=\{d_{1},d_{2},\ldots ,d_{n}\}\)(say) and \(D_{2}=\{d\in {\mathbb {F}}_{p^{m}} : d^{k}=1\}\) (\(k=p^{l}-1\) for a divisor l of m), first we define classical linear codes by where \(\textrm{Tr}^{m}_{e}\) denotes the trace function from \({\mathbb {F}}_{p^{m}}\) onto \({\mathbb {F}}_{p^{e}}\) and e is a divisor of m. Then we determine weight distribution of the code \(\overline{{\mathcal {C}}}_{D_{1}}\setminus {\mathcal {C}}_{D_{1}}\) and construct quantum codes from the codes \({\mathcal {C}}_{D_{1}}\) and \(\overline{{\mathcal {C}}}_{D_{1}}\) based on CSS code construction. Finally, we construct quantum code from the code \({\mathcal {C}}_{D_{2}}\) and show that the code obtained from \({\mathcal {C}}_{D_{2}}\) is MDS if and only if \(s=1,\) where \(s=\frac{m}{e}\).
$$\begin{aligned} {\mathcal {C}}_{D_{1}}= & {} \{(\textrm{Tr}_{e}^{m}(ad_{1}), \textrm{Tr}_{e}^{m}(ad_{2}),\ldots ,\textrm{Tr}_{e}^{m}(ad_{n})): a\in {\mathbb {F}}_{p^{m}}\}; \\ \overline{{\mathcal {C}}}_{D_{1}}= & {} \{( u+\textrm{Tr}_{e}^{m}(ad_{1}),u+\textrm{Tr}_{e}^{m}(ad_{2}),\ldots ,u+\textrm{Tr}_{e}^{m}(ad_{n})): a\in {\mathbb {F}}_{p^{m}}, u\in {\mathbb {F}}_{p^{e}}\}; \\ {\mathcal {C}}_{D_{2}}= & {} \{(\textrm{Tr}_{e}^{m}(ad_{1}),\ldots ,\textrm{Tr}_{e}^{m}(ad_{k}), u+\textrm{Tr}_{e}^{m}(ad_{1}),\ldots ,u+\textrm{Tr}_{e}^{m}(ad_{k})): \\ {}{} & {} a\in {\mathbb {F}}_{p^{m}}, u\in {\mathbb {F}}_{p^{e}}\}, \end{aligned}$$