<p>In this paper, we present a novel method for constructing designs and codes from the conjugacy classes and subgroups of finite groups. The resulting designs are point- and block-transitive and invariant under the group action. We apply our method to the sporadic simple group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ J_1 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>J</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> (Janko group) and its maximal subgroups, as well as to the group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ PSL_2(q) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>S</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ q $</EquationSource> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> </InlineEquation> is a power of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ 2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Our method generalises the ‘Key–Moori Method 2’ and uncovers several designs and codes not captured by that approach. Notably, it is effective for both maximal and non-maximal subgroups, broadening the scope of design construction. In particular, we provide specific examples of designs derived from non-maximal subgroups of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ PSL_2(q) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>S</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

A new method for constructing designs and codes from the conjugacy classes of groups

期刊论文
发表日期：

2025年12月20日

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Amin Saeidi,
Hadiseh Saydi,
Thekiso Seretlo

摘要

In this paper, we present a novel method for constructing designs and codes from the conjugacy classes and subgroups of finite groups. The resulting designs are point- and block-transitive and invariant under the group action. We apply our method to the sporadic simple group $ J_1 $ $J_{1}$ (Janko group) and its maximal subgroups, as well as to the group $ PSL_2(q) $ $P S L_{2} (q)$ , where $ q $ $q$ is a power of $ 2 $ $2$ . Our method generalises the ‘Key–Moori Method 2’ and uncovers several designs and codes not captured by that approach. Notably, it is effective for both maximal and non-maximal subgroups, broadening the scope of design construction. In particular, we provide specific examples of designs derived from non-maximal subgroups of $ PSL_2(q) $ $P S L_{2} (q)$ .