Skew-orthogonal steiner triple systems

Two Steiner triple systems, S₁=(V,ℬ︁₁) and S₂=(V,ℬ︁₂), are orthogonal (S₁ ⟂ S₂) if ℬ︁₁ ∩ ℬ︁₂=∅︁ and if {u,ν} ≠ {x,y}, uνw,xyw ∈ ℬ︁₁, uνs, xyt ∈ ℬ︁₂ then s ≠ t. The solution to the existence problem for orthogonal Steiner triple systems, (OSTS) was a major accomplishment in design theory. Two orthogonal triple systems are skew-orthogonal (SOSTS, written S₁∼S₂) if, in addition, we require uνw, xys ∈ ℬ︁₁ and uνt, xyw∈ ℬ︁₂ implies s ≠ t. Orthogonal triple systems are associated with a class of Room squares, with the skew orthogonal triple systems corresponding to skew Room squares. Also, SOSTS are related to separable weakly union-free TTS. SOSTS are much rarer than OSTS; for example SOSTS(ν) do not exist for ν=3,9,15. Furthermore, a fundamental construction for the earlier OSTS proofs when ν ≡ 3 (mod 6) cannot exist. In the case ν≡ 1 ( mod 6) we are able to show existence except possibly for 22 values, the largest of which is 1315. There are at least two non-isomorphic OSTS(19)s one of which is SOSTS(19) and the other not. A SOSTS(27) was found, implying the existence of SOSTS(ν) for ν ≡ 3 (mod 6) with finitely many possible exceptions.     

Disjoint finite partial steiner triple systems can be embedded in disjoint finite steiner triple systems

This paper shows that a pair of disjoint finite partial Steiner triple systems can be embedded in a pair of disjoint finite Steiner triple systems.     

5-chromatic steiner triple systems

We show that, up to an automorphism, there is a unique independent set in PG(5,2) that meets every hyperplane in 4 points or more. Using this result, we show that PG(5,2) is a 5-chromatic STS. Moreover, we construct a 5-chromatic STS(v) for every admissible v ≥ 127. © 1994 John Wiley & Sons, Inc.     

Embedding hypertrees into steiner triple systems

In this paper, we are interested in the following question: given an arbitrary Steiner triple system on vertices and any 3‐uniform hypertree on vertices, is it necessary that contains as a subgraph provided ? We show the answer is positive for a class of hypertrees and conjecture that the answer is always positive.     

Ternary codes of steiner triple systems

The code over a finite field F^q of a design ?? is the space spanned by the incidence vectors of the blocks. It is shown here that if ?? is a Steiner triple system on v points, and if the integer d is such that 3^d ≤ v < 3^d+1, then the ternary code C of ?? contains a subcode that can be shortened to the ternary generalized Reed-Muller code ?_F3(2(d ? 1),d) of length 3^d. If v = 3^d and d ≥ 2, then C_? ? ?_F3(1,d)? ? _F3(2(d ? 1),d) ? C. © 1994 John Wiley & Sons, Inc.     

A construction of disjoint steiner triple systems

A set of n ? 2 disjoint Steiner triple systems on n objects is constructed whenever n has the property that the order of 2 modulo n ? 2 is an odd number.     

Embedding partial steiner triple systems is NP-complete

It is known that any partial Steiner triple system of order υ can be embedded in a Steiner triple system of order w whenever w?4υ+1, and w≡1, 3 (mod 6); moreover, it is conjectured that the same is true whenever w?2υ+1. By way of contrast, it is proved that deciding whether a partial Steiner triple system of order υ can be embedded in a Steiner triple system of order w for any w?2υ?1 is NP-complete. In so doing, it is proved that deciding whether a partial commutative quasigroup can be completed to a commutative quasigroup is NP-complete.     

On the number of steiner triple systems

We obtain a new lower estimate for the number N(n) of nonisomorphic Steiner triple systems of order n: $$N(n) \geqslant n^{\frac{{n^2 }}{{12}} - O\left( {\frac{{n^2 }}{{logn}}} \right)} .$$ This makes it possible to show that log N(n) is of order n² log n.     

Generalized steiner triple systems with group size five

Generalized Steiner triple systems, GS(2, 3, n, g) are used to construct maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The existence of GS(2, 3, n, g) has been solved for g=2, 3, 4, 9. In this paper, by introducing a special kind of holey generalized Steiner triple systems (denoted by HGS(2, 3, (n, u), g)), singular indirect product (SIP) construction for GDDs is used to construct generalized Steiner systems. The numerical necessary conditions for the existence of a GS(2, 3, n, g) are shown to be sufficient for g=5.     

The spectrum of maximal partial steiner triple systems

The determination of the possible numbers of triples in a maximal partial triple system of order begun by Severn is completed. The result rests on the characterization of the maximum number of edges in a triangle-free, nonbipartite, antieulerian graph onn vertices.     

On intersections of pairs of steiner triple systems

Numerous articles exist in the literature concerning the intersection properties of collections of Steiner triple systems based on the same point set ([4], [5], [11], [12], [14], [15], [16], [19], [20]). In this paper we discuss several methods, first used by the authors in [7], for treating such problems. We apply these methods to reprove some known results and to furnish several new results.     

On large sets of disjoint steiner triple systems I

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, it is proved that if D(2 + n) = n, p is a prime number, p ≡ 7 (mod 8) or p? {5, 17, 19, 2}, and (p, n) ≠ (5, 1), then D(2 + pn) = pn.     

On large sets of disjoint steiner triple systems,V

In a previous paper (J. Combin. Theory Ser. A34 (1983), 156–182), to construct large sets of disjoint STS(3n)'s (i.e., LTS(3n)'s), a kind of combinatorial design, denoted by LD(n), where n is the order of design, was introduced and it was shown that if there exist both an LD(n) and an LTS(n + 2), then there exists an LTS(3n) also. In this paper, after having established some recursive theorems of LD(n), the following result was proved: If n is a positive integer such that n≡11 (mod 12), then there exists an LD(n), except possibly n ∈ {23, 47, 59, 83, 107, 167, 179, 227, 263, 299, 347, 383, 719, 767, 923, 1439}.     

On large sets of disjoint steiner triple systems,VI

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that D(v) = v ? 2 holds for all v ≡ 1, 3 (mod 6) (v>7), except possibly v = 141, 283, 501, 789, 1501, 2365.     

The use of hill-climbing to construct orthogonal steiner triple systems

In a related article, Colbourn, Gibbons, Mathon, Mullin, and Rosa [7] have shown that a pair of orthogonal Steiner triple systems exists for all v ≡ 1, 3 (mod 6), v ≥ 7 and v ≠ 9. This result is based on the construction of a finite set of pairs of orthogonal Steiner triple systems followed by the application of recursive constructions to settle the remaining undecided cases. In this article we report on the computational aspects of that investigation, and in particular the remarkable success of the hill-climbing method. © 1993 John Wiley & Sons, Inc.     

A theorem on the maximum number of disjoint steiner triple systems

We prove that D(2v + 1) ? v + 1 + D(v) for v > 3 where D(v) denotes the maximum number of pairwise disjoint Steiner triple systems of order v. Since D(v) ? v ? 2 it follows that for v > 3, D(2v + 1) = 2v ?1 whenever D(v) = v ? 2.