Search results for "Steiner triple system"
showing 4 items of 4 documents
On the additivity of block designs
2016
We show that symmetric block designs $${\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})$$D=(P,B) can be embedded in a suitable commutative group $${\mathfrak {G}}_{\mathcal {D}}$$GD in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of $${\mathrm {PG}}(d,2)$$PG(d,2) and $${\mathrm {AG}}(d,3)$$AG(d,3). In both cases, the blocks can be characterized as the only k-subsets of $$\mathcal {P}$$P whose elements sum to zero. It follows that the group of automorphisms of any such design $$\mathcal {D}$$D is the group of automorphisms of $${\mathfrak {G}}_\mathcal {D}$$GD that leave $$\mathcal {P}$$P in…
Additive Steiner triple systems
2014
A Steiner triple system is additive if it can be embedded in a commutative group in such a way that the sum of the three points in any given block is zero. In this paper we show that a Steiner triple system is additive if and only if it is the point-line design of either a projective space PG(d,2) over GF(2) or an affine space AG(d,3) over GF(3), for d ≥ 1. Our proof is based on algebraic arguments and on the combinatorial characterization of finite projective geometries in terms of Veblen points.
An algebraic representation of Steiner triple systems of order 13
2021
Abstract In this paper we construct an incidence structure isomorphic to a Steiner triple system of order 13 by defining a set B of twentysix vectors in the 13-dimensional vector space V = GF ( 5 ) 13 , with the property that there exist precisely thirteen 6-subsets of B whose elements sum up to zero in V , which can also be characterized as the intersections of B with thirteen linear hyperplanes of V .
Steiner Loops of Affine Type
2020
Steiner loops of affine type are associated to arbitrary Steiner triple systems. They behave to elementary abelian 3-groups as arbitrary Steiner Triple Systems behave to affine geometries over GF(3). We investigate algebraic and geometric properties of these loops often in connection to configurations. Steiner loops of affine type, as extensions of normal subloops by factor loops, are studied. We prove that the multiplication group of every Steiner loop of affine type with n elements is contained in the alternating group An and we give conditions for those loops having An as their multiplication groups (and hence for the loops being simple).