In this work, we first determine the generator polynomials of the Euclidean sums and hulls of cyclic codes over the commutative Frobenius ring <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4795_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_2\times (\mathbb {F}_2+v\mathbb {F}_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>v</mi> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4795_Article_IEq4.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>. Then, using CSS construction and quantum construction X of the Euclidean dual, we construct quantum error-correcting (QEC, for short) codes from the Euclidean sums and hulls of cyclic codes of length n over <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4795_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {F}_2\times (\mathbb {F}_2+v\mathbb {F}_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>v</mi> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Some new QEC codes are given in the sense that their parameters are different from all the previous constructions. We also construct entanglement-assisted quantum error-correcting (EAQEC, for short) codes from matrix product codes of the LCD codes of length n over this commutative Frobenius ring. To enrich the variety of available EAQEC codes, some new EAQEC codes are constructed to illustrate our results.

New QEC and EAQEC codes from Euclidean sums and hulls of cyclic codes over $\mathbb {F}_2\times (\mathbb {F}_2+\varvec{v}\mathbb {F}_2)$

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发表日期：

2025年6月16日

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Peng Hu,
Xiusheng Liu

摘要

In this work, we first determine the generator polynomials of the Euclidean sums and hulls of cyclic codes over the commutative Frobenius ring $\mathbb {F}_2\times (\mathbb {F}_2+v\mathbb {F}_2)$ $F_{2} \times (F_{2} + v F_{2})$ with $v^2=v$ $v^{2} = v$ . Then, using CSS construction and quantum construction X of the Euclidean dual, we construct quantum error-correcting (QEC, for short) codes from the Euclidean sums and hulls of cyclic codes of length n over $\mathbb {F}_2\times (\mathbb {F}_2+v\mathbb {F}_2)$ $F_{2} \times (F_{2} + v F_{2})$ . Some new QEC codes are given in the sense that their parameters are different from all the previous constructions. We also construct entanglement-assisted quantum error-correcting (EAQEC, for short) codes from matrix product codes of the LCD codes of length n over this commutative Frobenius ring. To enrich the variety of available EAQEC codes, some new EAQEC codes are constructed to illustrate our results.

New QEC and EAQEC codes from Euclidean sums and hulls of cyclic codes over \(\mathbb {F}_2\times (\mathbb {F}_2+\varvec{v}\mathbb {F}_2)\)

摘要