Examples of additive designs

In this paper we present some additive designs.

On 2-(n^2,2n,2n-1) designs with three intersection numbers

The simple incidence structure $${\mathcal{D}(\mathcal{A},2)}$$ , formed by the points and the unordered pairs of distinct parallel lines of a finite affine plane $${\mathcal{A}=(\mathcal{P}, \mathcal{L})}$$ of order n > 4, is a 2 --- (n 2,2n,2n---1) design with intersection numbers 0,4,n. In this paper, we show that the converse is true, when n ? 5 is an odd integer.

A note about 2-(v,5,lambda) designs

Uniqueness of AG3(4,2)

The simple incidence structure D(A, 2) formed by points and un-ordered pairs of distinct parallel lines of a finite affine plane A = (P,L) of order n > 2 is a 2 − (n^2, 2n, 2n − 1) design. If n = 3, D(A, 2) is the complementary design of A. If n = 4, D(A, 2) is isomorphic to the geometric design AG3(4, 2) (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a 2−(n^2, 2n, 2n−1) design to be of the form D(A, 2) for some finite affine plane A of order n > 4. As a consequence we obtain a characterization of small designs D(A, 2).

A note about additive designs

On the additivity of block designs

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…

2-(n2,2n,2n-1) designs obtained from affine planes

The simple incidence structure D(A, 2) formed by points and un- ordered pairs of distinct parallel lines of a finite affine plane A = (P,L) of order n > 2 is a 2 − (n^2, 2n, 2n − 1) design. If n = 3, D(A, 2) is the com- plementary design of A. If n = 4, D(A, 2) is isomorphic to the geometric design AG3(4, 2) (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a 2−(n^2, 2n, 2n−1) design to be of the form D(A, 2) for some finite affine plane A of order n > 4. As a consequence we obtain a characterization of small designs D(A, 2).

Some additive 2-(v,4,lambda) designs.

Additivity of affine designs

We show that any affine block design $$\mathcal{D}=(\mathcal{P},\mathcal{B})$$ is a subset of a suitable commutative group $${\mathfrak {G}}_\mathcal{D},$$ with the property that a k-subset of $$\mathcal{P}$$ is a block of $$\mathcal{D}$$ if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design $$\mathcal{D}$$ is the group of automorphisms of $${\mathfrak {G}}_\mathcal{D}$$ that leave $$\mathcal P$$ invariant. Whenever k is a prime p,  $${\mathfrak {G}}_\mathcal{D}$$ is an elementary abelian p-group.