An important family of quantum codes is the quantum maximum-distance-separable (MDS) codes. In this paper, we construct some new classes of quantum MDS codes by generalized Reed–Solomon (GRS) codes and Hermitian construction. In addition, the length n of the quantum MDS codes we constructed satisfies <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4788_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$n\equiv 0,1 ($</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≡</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">(</mo> </mrow> </math></EquationSource> </InlineEquation>mod<InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4788_Article_IEq10.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\,\frac{q\pm 1}{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mfrac> <mrow> <mi>q</mi> <mo>±</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which is different from previously known code lengths. At the same time, the quantum MDS codes we construct have length greater than <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4788_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$q+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and distance greater than <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4788_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$q/2+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, we can obtain a class of quantum MDS codes with distance q that has reached the maximum distance that can be constructed from GRS codes at that length. Finally, we can directly obtain many (MDS) entanglement-assisted quantum error correction codes (EAQECCs) with flexible parameters.

Quantum MDS codes with length $n\equiv 0,1($mod$\,\frac{q+1}{2})$ or $n\equiv 0,1($mod$\,\frac{q-1}{2})$

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发表日期：

2025年6月6日

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Ruhao Wan,
Mengchen Lian,
Shixin Zhu

摘要

An important family of quantum codes is the quantum maximum-distance-separable (MDS) codes. In this paper, we construct some new classes of quantum MDS codes by generalized Reed–Solomon (GRS) codes and Hermitian construction. In addition, the length n of the quantum MDS codes we constructed satisfies $n\equiv 0,1 ($ $n \equiv 0, 1 ($ mod $\,\frac{q\pm 1}{2})$ $\frac{q \pm 1}{2})$ , which is different from previously known code lengths. At the same time, the quantum MDS codes we construct have length greater than $q+1$ $q + 1$ and distance greater than $q/2+1$ $q / 2 + 1$ . In particular, we can obtain a class of quantum MDS codes with distance q that has reached the maximum distance that can be constructed from GRS codes at that length. Finally, we can directly obtain many (MDS) entanglement-assisted quantum error correction codes (EAQECCs) with flexible parameters.

Quantum MDS codes with length \(n\equiv 0,1(\)mod\(\,\frac{q+1}{2})\) or \(n\equiv 0,1(\)mod\(\,\frac{q-1}{2})\)

摘要