A design is G-additive, with G an abelian group, if its points are in G and each block is zero-sum in G. All the few known “manageable” additive Steiner 2-designs are <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{EA}(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>EA</mtext> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-additive for a suitable q, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{EA}(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>EA</mtext> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the elementary abelian group of order q. We present some general constructions for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{EA}(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>EA</mtext> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-additive Steiner 2-designs which unify the known ones and allow to find a few new ones: an <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{EA}(2^8)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>EA</mtext> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mn>8</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-additive 2-(52, 4, 1) design which is also resolvable, and three pairwise non-isomorphic <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{EA}(3^5)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>EA</mtext> <mo stretchy="false">(</mo> <msup> <mn>3</mn> <mn>5</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-additive 2-(121, 4, 1) designs, none of which is the point-line design of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{PG}(4,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>PG</mtext> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In the attempt to find also an <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{EA}(2^9)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>EA</mtext> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mn>9</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-additive 2-(511, 7, 1) design, we prove that a putative (subspace) 2-analog of a 2-(9, 3, 1) design cannot be cyclic.

$\textrm{EA}(q)$-additive Steiner 2-designs

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2026年5月9日

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Marco Buratti,
Mario Galici,
Alessandro Montinaro,
Anamari Nakić,
Alfred Wassermann

摘要

A design is G-additive, with G an abelian group, if its points are in G and each block is zero-sum in G. All the few known “manageable” additive Steiner 2-designs are $\textrm{EA}(q)$ $EA (q)$ -additive for a suitable q, where $\textrm{EA}(q)$ $EA (q)$ is the elementary abelian group of order q. We present some general constructions for $\textrm{EA}(q)$ $EA (q)$ -additive Steiner 2-designs which unify the known ones and allow to find a few new ones: an $\textrm{EA}(2^8)$ $EA (2^{8})$ -additive 2-(52, 4, 1) design which is also resolvable, and three pairwise non-isomorphic $\textrm{EA}(3^5)$ $EA (3^{5})$ -additive 2-(121, 4, 1) designs, none of which is the point-line design of $\textrm{PG}(4,3)$ $PG (4, 3)$ . In the attempt to find also an $\textrm{EA}(2^9)$ $EA (2^{9})$ -additive 2-(511, 7, 1) design, we prove that a putative (subspace) 2-analog of a 2-(9, 3, 1) design cannot be cyclic.

\(\textrm{EA}(q)\)-additive Steiner 2-designs

摘要