Search results for "Incidence structure"
showing 3 items of 3 documents
On 2-(n^2,2n,2n-1) designs with three intersection numbers
2007
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.
Über die Schnittzahlen mehrfach balancierter blockpläne
1991
Abstract For a finite incidence structure D with a set X of blocks let [ X ] be the number of points common with all blocks contained in X . We define the functions M(t)(B1,…; B1)=ΣB [B1, B]…[B1,B], and, for every partition ϖ = ϖ1,…,ϖ1) of t, the function Mϖ(B1,…,B1) = Σ Πm [Bi | i ϵ Rm], sum over all decompositions {l, …, t} = R1, ⊃ … ⊃ Rl, |Rm| = ϖm. We show: If D is t-fold balanced, then M(t) = Σϖ cϖMϖ, where the, coefficients cϖ are linear combinations of the parameters b1,…,bt, the constant numbers of blocks through any l,…, t distinct points. Conversely, if the rank of the b × b-matrix ([B, B∗])B,B∗ is equal to the number ν of points and M(t) is a rational linear combination of the fu…
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 .