Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="186" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q^{2}}\mathcal {R}=\mathbb {F}_{q^{2}} \times (\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mi mathvariant="script">R</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mo>+</mo> <mi>v</mi> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where q is an odd prime power and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$v^{2}=v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> <mo>=</mo> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we discuss the properties of linear codes and u-constacyclic codes over <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q^{2}}\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mi mathvariant="script">R</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">$u=(u_1,u_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq8.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$u_1\in \mathbb {F}_{q^2}^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </InlineMediaObject> <EquationSource Format="TEX">$u_2=\varepsilon (1-2v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>ε</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq10.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon \in \mathbb {F}_{q^2}^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation>. Besides, a Gray map from <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq11.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q^{2}}^{m}\times \mathcal {R}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mi>m</mi> </msubsup> <mo>×</mo> <msup> <mrow> <mi mathvariant="script">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq12.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q^{2}}^{m+2n}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">F</mi> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>n</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation> is defined, and the Gray images of linear codes and the separable <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q^{2}}\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mi mathvariant="script">R</mi> </mrow> </math></EquationSource> </InlineEquation>-u-constacyclic codes are studied. According to the Gray images of the separable <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4737_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_{q^{2}}\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mi mathvariant="script">R</mi> </mrow> </math></EquationSource> </InlineEquation>-u-constacyclic codes, some new quantum codes are obtained. Compared with the known ones, our codes have better parameters.

Constacyclic codes over $\mathbb {F}_{q^{2}} \times (\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}})$ and their applications in constructing new quantum codes

期刊论文
发表日期：

2025年4月28日

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Liqi Wang,
Xinxin Zhang,
Shixin Zhu

摘要

Let $\mathbb {F}_{q^{2}}\mathcal {R}=\mathbb {F}_{q^{2}} \times (\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}})$ $F_{q^{2}} R = F_{q^{2}} \times (F_{q^{2}} + v F_{q^{2}})$ , where q is an odd prime power and $v^{2}=v$ $v^{2} = v$ . In this paper, we discuss the properties of linear codes and u-constacyclic codes over $\mathbb {F}_{q^{2}}\mathcal {R}$ $F_{q^{2}} R$ , where $u=(u_1,u_2)$ $u = (u_{1}, u_{2})$ , $u_1\in \mathbb {F}_{q^2}^*$ $u_{1} \in F_{q^{2}}^{*}$ , $u_2=\varepsilon (1-2v)$ $u_{2} = ε (1 - 2 v)$ , and $\varepsilon \in \mathbb {F}_{q^2}^*$ $ε \in F_{q^{2}}^{*}$ . Besides, a Gray map from $\mathbb {F}_{q^{2}}^{m}\times \mathcal {R}^{n}$ $F_{q^{2}}^{m} \times R^{n}$ to $\mathbb {F}_{q^{2}}^{m+2n}$ $F_{q^{2}}^{m + 2 n}$ is defined, and the Gray images of linear codes and the separable $\mathbb {F}_{q^{2}}\mathcal {R}$ $F_{q^{2}} R$ -u-constacyclic codes are studied. According to the Gray images of the separable $\mathbb {F}_{q^{2}}\mathcal {R}$ $F_{q^{2}} R$ -u-constacyclic codes, some new quantum codes are obtained. Compared with the known ones, our codes have better parameters.

Constacyclic codes over \(\mathbb {F}_{q^{2}} \times (\mathbb {F}_{q^{2}}+v\mathbb {F}_{q^{2}})\) and their applications in constructing new quantum codes

摘要