Super-regular Steiner 2-designs

A design is additive under an abelian group G (briefly, G-additive) if, up to isomorphism, its point set contained in and the elements of each block sum zero. The only known Steiner 2-designs that are G-additive for some have size which either a prime power or plus one. Indeed they point-line designs affine spaces AG(n,q), projective planes PG(2,q), PG(n,2) sporadic example 2-(8191,7,1) design. In attempt find new examples, possibly with neither nor one, we look strictly (the exactly G) G-regular (any translate any as well) at same time. These will be called “G-super-regular”. Our main result there infinitely many values v exists super-regular, therefore additive, 2-(v,k,1) whenever k singly even form 2n3≥12. case k≡2 (mod 4) genuine exception whereas k=2n3≥12 moment possible exception. We also super-regular 2-(pn,p,1) p∈{5,7} n≥3 not isomorphic AG(n,p).