In this paper, we study non-complete G-block-transitive 3-<InlineEquation ID="IEq4"> <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 with small values of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$T \le G \le \textrm{Aut}(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>≤</mo> <mi>G</mi> <mo>≤</mo> <mtext>Aut</mtext> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and T is a sporadic simple group. When G acts primitively on the point set and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda \le 10,000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≤</mo> <mn>10</mn> <mo>,</mo> <mn>000</mn> </mrow> </math></EquationSource> </InlineEquation>, we obtain a list of possible parameter tuples <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(G,v,k,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and explicitly construct several new designs for the Mathieu group <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{M}_{22}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>M</mtext> <mn>22</mn> </msub> </math></EquationSource> </InlineEquation> and the Higman-Sims group <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\textrm{HS}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>HS</mtext> </math></EquationSource> </InlineEquation>. Under the condition <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\lambda \le 123$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≤</mo> <mn>123</mn> </mrow> </math></EquationSource> </InlineEquation>, we obtain a full classification: up to isomorphism, there are exactly 49 such designs. Each of these 49 designs arises from a 3-homogeneous action of one of the five Mathieu groups.

Block-transitive 3-designs with small $\lambda $ on sporadic simple groups

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2026年7月28日

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Ting Lan,
Weijun Liu,
Fu-Gang Yin

摘要

In this paper, we study non-complete G-block-transitive 3- $(v,k,\lambda )$ $(v, k, λ)$ designs with small values of $\lambda $ $λ$ , where $T \le G \le \textrm{Aut}(T)$ $T \leq G \leq Aut (T)$ and T is a sporadic simple group. When G acts primitively on the point set and $\lambda \le 10,000$ $λ \leq 10, 000$ , we obtain a list of possible parameter tuples $(G,v,k,\lambda )$ $(G, v, k, λ)$ and explicitly construct several new designs for the Mathieu group $\textrm{M}_{22}$ $M_{22}$ and the Higman-Sims group $\textrm{HS}$ $HS$ . Under the condition $\lambda \le 123$ $λ \leq 123$ , we obtain a full classification: up to isomorphism, there are exactly 49 such designs. Each of these 49 designs arises from a 3-homogeneous action of one of the five Mathieu groups.

Block-transitive 3-designs with small \(\lambda \) on sporadic simple groups

摘要