On sparse countably infinite Steiner triple systems (original) (raw)

Perfect countably infinite Steiner triple systems

We use a free construction to prove the existence of perfect Steiner triple systems on a countably infinite point set. We use a specific countably infinite family of partial Steiner triple systems to start the construction, thus yielding 2 ℵ0 non-isomorphic perfect systems.

Further 6-sparse Steiner Triple Systems

Graphs and Combinatorics, 2009

We give a construction that produces 6-sparse Steiner triple systems of order v for all sufficiently large v of the form 3p, p prime and p ≡ 3 (mod 4). We also give a complete list of all 429 6-sparse systems with v < 10000 produced by this construction.

On a More General Characterisation of Steiner Systems

Designs, Codes and Cryptography, 2005

We introduce [k,d]-sparse geometries of cardinality n, which are natural generalizations of partial Steiner systems PS(t,k;n), with d=2(k−t+1). We will verify whether Steiner systems are characterised in the following way. (*) Let Gamma=(mathcalP,mathcalB)\Gamma=(\mathcal{P},\mathcal{B})Gamma=(mathcalP,mathcalB) be a [k,2(k−t+1)]-sparse geometry of cardinality n, with fracn+12>k>t>1\frac{n+1}{2} \> k \> t \> 1fracn+12>k>t>1 . If ∣mathcalB∣genchooset/kchooset|\mathcal{B}| \ge {n \choose t}/{k \choose t}∣mathcalB∣genchooset/kchooset , then Γ is a S(t,k;n). If (*) holds for fixed parameters t, k and n, then we say S(t,k;n) satisfies, or has, characterisation (*). We could not prove (*) in general, but we prove the Theorems 1, 2, 3 and 4, which state conditions under which (*) is satisfied. Moreover, we verify characterisation (*) for every Steiner system appearing in list of the sporadic Steiner systems of small cardinality, and the list of infinite series of Steiner systems, both mentioned in the latest edition of the book ‘Design Theory’ by T. Beth, D. Jungnickel and H. Lenz. As an interesting application, one can use these results to build (almost) maximal binary codes in the following way. Every [k,d]-sparse geometry is associated with a [k,d]-sparse binary code of the same size (let mathcalP=p1,ldots,pn\mathcal{P} = \{ p_1, \ldots, p_n \}mathcalP=p1,ldots,pn and link every block BinmathcalBB \in \mathcal{B}BinmathcalB with the code word (ci)1leilen(c_i)_{1 \le i \le n}(ci)1leilen where c i =1 if and only if the point p i is a member of B), so one can construct maximal [k,d]-sparse binary codes using (partial) Steiner systems. These [k,d]-sparse codes can then be used as building bricks for binary codes having a bigger variety of weights (the weight of a code word is the sum of its entries).

A Construction of Almost Steiner Systems

Journal of Combinatorial Designs, 2013

Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k-uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.

On the Szamkolowicz-Doyen Classification of Steiner Triple Systems

Proceedings of the London Mathematical Society, 1977

1. Introduction A Steiner triple system (8, stf) is a finite set 8 together with a family s/ of triples (subsets of cardinality 3) such that each pair of elements of S is contained in exactly one triple. The elements of 8 are called points, the elements of s/ are called lines, and three points which are not on a line are a triangle. The cardinal of 8, which we denote by 18 |, is the order of the Steiner triple system (S,s/). It is well known [8] that a necessary and sufficient condition for the existence of a Steiner triple system of order n > 0 is that n = 1 or 3 (mod 6). Trivial Steiner triple systems occur when S = stf -0, when 8 consists of one point and s& = 0, and when 8 consists of three points and $2 consists of one line. A subsystem of a Steiner triple system (S, stf) is a Steiner triple system (8', &4') such that 8' <= 8 and s#' <= s0. It is easy to see that if (8 V J^) , ..., (8 r ,s0 r ) are subsystems of a Steiner triple system then so is (8 x n... n8 r ,s^n... ns/ r ). It follows that given a set P s S of points, there is a unique minimal subsystem containing P. This subsystem is said to be generated by P.

Properties of the Steiner Triple Systems of Order 19

The Electronic Journal of Combinatorics, 2010

Properties of the 11$\,$084$\,$874$\,$829 Steiner triple systems of order 19 are examined. In particular, there is exactly one 5-sparse, but no 6-sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number 3; all have chromatic index 10, except for 4$\,$075 designs with chromatic index 11 and two with chromatic index 12; all are 3-resolvable; and there are exactly two 3-existentially closed STS(19).

Some Remarks on Steiner Systems

2003

The main purpose of this paper is to introduce Steiner systems obtained from the finite classical generalized hexagons of order q. We show that we can take the blocks of the Steiner systems amongst the lines and the traces of the hexagon, and we prove some facts about the automorphism groups. Also, we make a remark concerning the geometric construction of a known class (KW) of Steiner systems and we deduce some properties of the automorphism group.

Line-closed subsets of Steiner triple systems and classical linear spaces

Journal of Statistical Planning and Inference, 1996

A proper non-empty subset C of the points of a linear space S = (P; L) is called line-closed if any two intersecting lines of S , each meeting C at least twice, have their intersection in C. We show that when every line has k points and every point lies on r lines the maximum size for such sets is r + k ? 2. In addition it is shown that this cannot happen for projective spaces PG(n; q) unless q = 2, nor can it be obtained for a ne spaces AG(n; q) unless n = 2 and q = 3. However, for all odd values of r there exist Steiner triple systems having such maximum line-closed subsets.

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