Quantum codes from trace codes

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

$$\begin{aligned} {\mathcal {C}}_{D_{1}}= & {} \{(\textrm{Tr}_{e}^{m}(ad_{1}), \textrm{Tr}_{e}^{m}(ad_{2}),\ldots ,\textrm{Tr}_{e}^{m}(ad_{n})): a\in {\mathbb {F}}_{p^{m}}\}; \\ \overline{{\mathcal {C}}}_{D_{1}}= & {} \{( u+\textrm{Tr}_{e}^{m}(ad_{1}),u+\textrm{Tr}_{e}^{m}(ad_{2}),\ldots ,u+\textrm{Tr}_{e}^{m}(ad_{n})): a\in {\mathbb {F}}_{p^{m}}, u\in {\mathbb {F}}_{p^{e}}\}; \\ {\mathcal {C}}_{D_{2}}= & {} \{(\textrm{Tr}_{e}^{m}(ad_{1}),\ldots ,\textrm{Tr}_{e}^{m}(ad_{k}), u+\textrm{Tr}_{e}^{m}(ad_{1}),\ldots ,u+\textrm{Tr}_{e}^{m}(ad_{k})): \\ {}{} & {} a\in {\mathbb {F}}_{p^{m}}, u\in {\mathbb {F}}_{p^{e}}\}, \end{aligned}$$

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 $s=\frac{m}{e}$.