Let p be a prime number, m be a positive integer, and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1624_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$q=p^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. For any fixed locality r such that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1624_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$p\not \mid r(r+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∤</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we construct infinite families of locally recoverable codes with availabilty of nodes lower bounded by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1624_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$q/r!+O(\sqrt{q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo stretchy="false">/</mo> <mi>r</mi> <mo>!</mo> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msqrt> <mi>q</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and number of locality sets equal to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1624_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="150" /> </InlineMediaObject> <EquationSource Format="TEX">$q^2/(r+1)!+O(q^{3/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>!</mo> <mo>+</mo> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Constructions of locally recoverable codes with large availability

期刊论文
发表日期：

2025年4月5日

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Giacomo Micheli,
Vincenzo Pallozzi Lavorante,
Abhi Shukul,
Noah Smith

摘要

Let p be a prime number, m be a positive integer, and $q=p^m$ $q = p^{m}$ . For any fixed locality r such that $p\not \mid r(r+1)$ $p ∤ r (r + 1)$ , we construct infinite families of locally recoverable codes with availabilty of nodes lower bounded by $q/r!+O(\sqrt{q})$ $q / r! + O (\sqrt{q})$ and number of locality sets equal to $q^2/(r+1)!+O(q^{3/2})$ $q^{2} / (r + 1)! + O (q^{3 / 2})$ .