EA(q)-additive Steiner 2-designs

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 EA(q) EA ( q ) -additive for a suitable q , where EA(q) EA ( q ) is the elementary abelian group of order q . We present some general constructions for EA(q) EA ( q ) -additive Steiner 2-designs which unify the known ones and allow to find a few new ones: an EA(2⁸) EA ( 2 8 ) -additive 2-(52, 4, 1) design which is also resolvable, and three pairwise non-isomorphic EA(3⁵) EA ( 3 5 ) -additive 2-(121, 4, 1) designs, none of which is the point-line design of PG(4,3) PG ( 4 , 3 ) . In the attempt to find also an EA(2⁹) 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.