Let <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {R}_q=\mathbb {F}_q\times (\mathbb {F}_q+v\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mi>q</mi> </msub> <mo>=</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>+</mo> <mi>v</mi> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq8"> <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> and q be an odd prime power. Firstly, we give two Gray maps from <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {R}_q^n$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">R</mi> <mi>q</mi> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {F}_q^{3n}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">F</mi> <mi>q</mi> <mrow> <mn>3</mn> <mi>n</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. Next, we provide the structure of all <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$(1,1-2v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-constacyclic codes and their dual codes with a length n over <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathcal {R}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>. By means of these structures, we give also the structure of all Euclidean hulls and sums of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$(1,1-2v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-constacyclic codes of length n over <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathcal {R}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>. Finally, using Steane’s construction and quantum construction X of the Euclidean dual, we provide two methods for constructing quantum error-correcting (QEC, for short) codes via the Euclidean sums and hulls of <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$(1,1-2v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-constacyclic codes and linear codes of length n over <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\mathcal {R}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>. To enrich the variety of available QEC codes, some new QEC codes are constructed to illustrate our results.

$(1,1-2v)$-constacyclic codes over $\mathbb {F}_q\times (\mathbb {F}_q+v\mathbb {F}_q)$ and their applications to QEC codes

期刊论文
发表日期：

2026年7月27日

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Xiusheng Liu,
Jie Liu

摘要

Let $\mathcal {R}_q=\mathbb {F}_q\times (\mathbb {F}_q+v\mathbb {F}_q)$ $R_{q} = F_{q} \times (F_{q} + v F_{q})$ with $v^2=v$ $v^{2} = v$ and q be an odd prime power. Firstly, we give two Gray maps from $\mathcal {R}_q^n$ $R_{q}^{n}$ to $\mathbb {F}_q^{3n}$ $F_{q}^{3 n}$ . Next, we provide the structure of all $(1,1-2v)$ $(1, 1 - 2 v)$ -constacyclic codes and their dual codes with a length n over $\mathcal {R}_q$ $R_{q}$ . By means of these structures, we give also the structure of all Euclidean hulls and sums of $(1,1-2v)$ $(1, 1 - 2 v)$ -constacyclic codes of length n over $\mathcal {R}_q$ $R_{q}$ . Finally, using Steane’s construction and quantum construction X of the Euclidean dual, we provide two methods for constructing quantum error-correcting (QEC, for short) codes via the Euclidean sums and hulls of $(1,1-2v)$ $(1, 1 - 2 v)$ -constacyclic codes and linear codes of length n over $\mathcal {R}_q$ $R_{q}$ . To enrich the variety of available QEC codes, some new QEC codes are constructed to illustrate our results.

\((1,1-2v)\)-constacyclic codes over \(\mathbb {F}_q\times (\mathbb {F}_q+v\mathbb {F}_q)\) and their applications to QEC codes

摘要