This paper classifies symmetric <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2341_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$(v,k,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> designs <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2341_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {D}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation> with v odd whose automorphism group G is block-transitive and of almost simple type with alternating socle. Finally we prove that if <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2341_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$6\le \lambda \le 10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>6</mn> <mo>≤</mo> <mi>λ</mi> <mo>≤</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2341_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {D}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation> is a unique symmetric (35, 17, 8) design or its complement and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2341_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$G=\mathrm {A_{m}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">m</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2341_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathrm {S_{m}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">m</mi> </msub> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2341_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$(m=7, 8)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>=</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Block-Transitive Symmetric $(v,k,\lambda )$ Designs with $\lambda $ at Most 10 and Alternating Socle

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发表日期：

2025年1月2日

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Jiakun Chen,
Xiaohong Zhang

摘要

This paper classifies symmetric $(v,k,\lambda )$ $(v, k, λ)$ designs ${\mathcal {D}}$ $D$ with v odd whose automorphism group G is block-transitive and of almost simple type with alternating socle. Finally we prove that if $6\le \lambda \le 10$ $6 \leq λ \leq 10$ , then ${\mathcal {D}}$ $D$ is a unique symmetric (35, 17, 8) design or its complement and $G=\mathrm {A_{m}}$ $G = A_{m}$ , $\mathrm {S_{m}}$ $S_{m}$ $(m=7, 8)$ $(m = 7, 8)$ .

Block-Transitive Symmetric \((v,k,\lambda )\) Designs with \(\lambda \) at Most 10 and Alternating Socle

摘要