A significant number of quantum maximal-distance-separable (MDS) codes have been successfully developed via constacyclic codes through the application of the Hermitian construction method. While existing quantum error-correcting codes (QECCs) often feature lengths that are divisors of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4900_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$q^w-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>q</mi> <mi>w</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, with q being a prime power and w representing a positive even number, this work introduces a novel family of quantum MDS codes with lengths dividing <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4900_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$q^w-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>q</mi> <mi>w</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where w is instead an odd prime number. These codes have lengths exceeding <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4900_Article_IEq5.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 minimum distances larger than <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4900_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{q}{2}+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mi>q</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Additionally, two specific quantum MDS codes for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4900_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$w=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> are presented.

A family of new quantum MDS codes with lengths dividing $q^w-1$

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发表日期：

2025年8月19日

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Fu-Yu Gu,
Rui Wang,
Yi Li,
Ming-Qiang Bai

摘要

A significant number of quantum maximal-distance-separable (MDS) codes have been successfully developed via constacyclic codes through the application of the Hermitian construction method. While existing quantum error-correcting codes (QECCs) often feature lengths that are divisors of $q^w-1$ $q^{w} - 1$ , with q being a prime power and w representing a positive even number, this work introduces a novel family of quantum MDS codes with lengths dividing $q^w-1$ $q^{w} - 1$ , where w is instead an odd prime number. These codes have lengths exceeding $q + 1$ $q + 1$ and minimum distances larger than $\frac{q}{2}+1$ $\frac{q}{2} + 1$ . Additionally, two specific quantum MDS codes for $w=3$ $w = 3$ are presented.

A family of new quantum MDS codes with lengths dividing \(q^w-1\)

摘要