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

Abstract 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 $$\textrm{EA}(q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>EA</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -additive for a suitable q , where $$\textrm{EA}(q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>EA</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is the elementary abelian group of order q . We present some general constructions for $$\textrm{EA}(q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>EA</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -additive Steiner 2-designs which unify the known ones and allow to find a few new ones: an $$\textrm{EA}(2^8)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>EA</mml:mtext> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>8</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -additive 2-(52, 4, 1) design which is also resolvable, and three pairwise non-isomorphic $$\textrm{EA}(3^5)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>EA</mml:mtext> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>3</mml:mn> <mml:mn>5</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -additive 2-(121, 4, 1) designs, none of which is the point-line design of $$\textrm{PG}(4,3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>PG</mml:mtext> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In the attempt to find also an $$\textrm{EA}(2^9)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>EA</mml:mtext> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>9</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -additive 2-(511, 7, 1) design, we prove that a putative (subspace) 2-analog of a 2-(9, 3, 1) design cannot be cyclic.