共查询到20条相似文献,搜索用时 0 毫秒
1.
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. 相似文献
2.
Charles C Lindner 《Journal of Combinatorial Theory, Series A》1975,18(1):126-129
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. 相似文献
3.
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. 相似文献
4.
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. 相似文献
5.
The code over a finite field Fq 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 3d ≤ v < 3d+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 3d. If v = 3d and d ≥ 2, then C? ? ?F3(1,d)? ? F3(2(d ? 1),d) ? C. © 1994 John Wiley & Sons, Inc. 相似文献
6.
7.
Donald Quiring 《Journal of Combinatorial Theory, Series A》1979,27(3):407-408
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. 相似文献
8.
Charles J Colbourn 《Journal of Combinatorial Theory, Series A》1983,35(1):100-105
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. 相似文献
9.
V. E. Alekseev 《Mathematical Notes》1974,15(5):461-464
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. 相似文献
10.
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. 相似文献
11.
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. 相似文献
12.
《Indagationes Mathematicae (Proceedings)》1977,80(2):89-100
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. 相似文献
13.
Lu Jia-Xi 《Journal of Combinatorial Theory, Series A》1983,34(2):140-146
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. 相似文献
14.
15.
16.
17.
Lu Jia-Xi 《Journal of Combinatorial Theory, Series A》1984,37(2):164-188
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}. 相似文献
18.
Lu Jia-Xi 《Journal of Combinatorial Theory, Series A》1984,37(2):189-192
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. 相似文献
19.
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. 相似文献
20.
Alexander Rosa 《Journal of Combinatorial Theory, Series A》1975,18(3):305-312
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. 相似文献