Luc Teirlinck

Some new 2-resolvable Steiner quadruple systems

Zaicev, Zinoviev and Semakov [12] and, independently, Baker [1], constructed 2-resolvableS(3, 4, 4n) for all ℕ. However, no 2-resolvableS(3, 4,v),v≥4, were known for any other value ofv. In this paper, we construct infinite classes of 2-resolvableS(3, 4,v) for values ofv that are not a power of 4. In particular, we construct a 2-resolvableS(3, 4, 100).

On the maximum number of disjoint Steiner triple systems

Let D(v) be the maximum number of pairwise disjoint Steiner triple systems of order v. We prove that D(3v)>=2v+D(v) for every v = 1 or 3 (mod 6), v>=3. As a corollary, we have D(3^n)=3^n-2 for every n>=1.

Locally trivial t -designs and t -designs without repeated blocks

We simplify our construction [12] of non-trivial t-designs without repeated blocks for arbitrary t. We survey known results on partitions of the set of all (t + l)-subsets of a u-set into S(λ; t, t + I, μ) for the smallest λ allowed by the obvious necessary conditions. We also obtain some new results on this problem. In particular, we construct such partitions for t = 4 and k = 60 whenever ν = 60u + 4, u a positive integer with gcd(u, 60) = I or 2. Sixty is the smallest possible λ for such ν.

A Construction for Authentication/secrecy Codes from 3-homogeneous Permutation Groups

In this paper, we construct codes which provide both secrecy and authentication using 3-homogeneous groups. We construct an infinite family of codes which provide perfect secrecy even if the same encoding rule is used three times in succession; and provide optimal protection against deception by an opponent who observes up to two authentic messages and then substitutes a message of his own choosing.

The construction of large sets of indempotent quasigroups

The maximum number of idempotent quasigroups of order n which pairwise agree on the main diagonal only is n − 2. Such a collection is called a large set of idempotent quasigroups of order n. The main result in this paper is the construction of a large set of idempotent quasigroups of order n for every n ⩾ 3 except n = 6, for which no such collection exists, and n = 14 and 62. Additionally, the known spectrum for large sets of Mendelsohn quasigroups is improved.

The existence of reverse Steiner triple systems

A Steiner triple system of order v is called reverse if its automorphism group contains an involution fixing only one point. We show that such a system exists if and only if v = 1,3,9 or 19 (mod 24).

Combinatorial properties of planar spaces and embeddability

Abstract Let S be a planar space in which all planes V are finite and contain N(V) + 1 lines through each point p ϵ V. Our main result states that, if |L| ⩾ (2N(V) + 5) 3 for any plane V and any line L of V, then N(V) is a constant N and S is embeddable in a projective space. This will be a consequence of several other, more general results, giving sufficient conditions for a planar space S to be locally projective and/or to satisfy the bundle property.

Factorization Properties for Isometries of Matroids into Projective Spaces

Let k, n ∈ ℕ, k ⩾ 2, n ⩾ 2 and let T be a class of desarguesian projective spaces, containing at least one projective space of order k and dimension n. We define μ (T, n, k) as the smallest m ∈ ℕ such that for every isometry α : V → P where V is a matroid of dimension n, |V|⩾ m, and P ∈ T has order k and dimension n, and for every isometry β : V → P′ ∈ T, there is a unique isometry γ : P → P′ such that β = γ ∘ α. We find lower and upper bounds for μ (T, n, k).

The Geometry of Subspaces of an S(λ;2,3,v)

We define some linear spaces on the set of all proper subspaces of a triple system S(γ2,3,v). The connected components of these linear spaces are projective spaces of order 2 and punctured projective spaces of order 3, i.e. projective spaces of order 3 from which a point has been deleted. We show how these connected components can be used to find affine and projective factors in S(γ2,3,v).