<p>Let <i>q</i> be a prime power. Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_834_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be a divisor of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_834_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </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 <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_834_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be a divisor of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_834_Article_IEq4.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>. Under certain conditions we prove that there exists an MDS stabilizer quantum code with length <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_834_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$n=\lambda \tau \sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mi>λ</mi> <mi>τ</mi> <mi>σ</mi> </mrow> </math></EquationSource> </InlineEquation> for a range of values of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_834_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>. This is a flexible construction, which includes previously known families and also new MDS parameters not known before.</p>

New quantum MDS codes with flexible parameters from Hermitian self-orthogonal GRS codes

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2025年8月13日

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Oisin Campion,
Fernando Hernando,
Gary McGuire

摘要

Let q be a prime power. Let $\lambda >1$ $λ > 1$ be a divisor of $q-1$ $q - 1$ and $\tau >1$ $τ > 1$ be a divisor of $q+1$ $q + 1$ . Under certain conditions we prove that there exists an MDS stabilizer quantum code with length $n=\lambda \tau \sigma $ $n = λ τ σ$ for a range of values of $\sigma $ $σ$ . This is a flexible construction, which includes previously known families and also new MDS parameters not known before.