Caps and Colouring Steiner Triple Systems

Hill [6] showed that the largest cap in PG(5,3) has cardinality 56. Using this cap it is easy to construct a cap of cardinality 45 in AG(5,3). Here we show that the size of a cap in AG(5,3) is bounded above by 48. We also give an example of three disjoint 45-caps in AG(5,3). Using these two results we are able to prove that the Steiner triple system AG(5,3) is 6-chromatic, and so we exhibit the first specific example of a 6-chromatic Steiner triple system.     

On the chromatic numbers of Steiner triple systems

Geometric properties are used to determine the chromatic number of AG(4, 3) and to derive some important facts on the chromatic number of PG(n, 2). It is also shown that a 4-chromatic STS(v) exists for every admissible order v ≥ 21. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 1–10, 1999     

On Large Sets of v−1 L-Intersecting Steiner Triple Systems of Order v

This paper presents four new recursive constructions for large sets of v–1 STS(v). These facilitate the production of several new infinite families of such large sets. In particular, we obtain for each n2 a large set of 3 ⁿ –1 STS (3 ⁿ ) whose systems intersect in 0 or 3 blocks.     

Nilpotent Steiner loops of class 2

We describe Steiner loops of nilpotency class 2 and establish the classification of finite 3-generated nilpotent Steiner loops of nilpotency class 2.     

Indecomposable large sets of Steiner triple systems with indices 5, 6

A family ( X, B1 ), (X, B2 ), . . . , (X, Bq ) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTS λ (v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTS λ (v) if there does not exist an LSTS λ'(v) contained in the collection for any λ' λ. In this paper, we show that for λ = 5, 6, there is an IDLSTS λ (v) for v ≡ 1 or 3 (mod 6) with the exception IDLSTS6 (7).     

组合设计的大集

组合设计中的大集问题有着悠久的历史和广泛的应用．由于它的难度，长期进展很慢．近二十多年来，在一些新的方法和手段的推动下，大集研究呈现了很好的态势．本文力图对几类主要组合设计大集的概念和研究进展给予概要介绍，以期引起更多的关注。     

Comment on “The Steiner number of a graph” by G. Chartrand and P. Zhang: [Discrete Mathematics 242 (2002) 41–54]

We show by counterexample that one of the main results in the paper “The Steiner number of a graph” by Chartrand and Zhang (Disc. Math. 242 (2002) 41–54) does not hold. To be more precise, we prove both that not every Steiner set is a geodetic set and that there are connected graphs whose Steiner number is strictly lower than its geodetic number.     

A tripling construction for overlarge sets of KTS

An overlarge set of , denoted by , is a collection {(X?{x},B_x):x∈X}, where X is a (v+1)-set, each (X?{x},B_x) is a and {B_x:x∈X} forms a partition of all triples on X. In this paper, we give a tripling construction for overlarge sets of KTS. Our main result is that: If there exists an with a special property, then there exists an . It is obtained that there exists an for u=2²ⁿ⁻¹−1 or u=qⁿ, where prime power q≡7 (mod 12) and m≥0,n≥1.     

Further results on overlarge sets of Kirkman triple systems

In this paper, we introduce a new concept -- overlarge sets of generalized Kirkman systems （OLGKS）, research the relation between it and OLKTS, and obtain some new results for OLKTS. The main conclusion is： If there exist both an OLKF（6^k） and a 3-OLGKS（6^k-1,4） for all k ∈{6,7,...,40}／{8,17,21,22,25,26}, then there exists an OLKTS（v） for any v ≡ 3 （mod 6）, v ≠ 21. As well, we obtain the following result： There exists an OLKTS（6u ＋ 3） for u = 2^2n-1 - 1, 7^n, 31^n, 127^n, 4^r25^s, where n ≥ 1,r＋s≥ 1.     

Maximum Kirkman Signal Sets for Synchronous Uni-Polar Multi-UserCommunication Systems

A unipolar signalingsystem transmits using intensity or amplitude in multiple dimensions.Typical examples arise in optical transmission or radio communicationusing MT-MFSK as both the signaling and the modulation technique.There are dimensions which represent pulses ortones. Each codeword consists of a selection of kof these tones with unit intensity. Each user is assigned mof these binary codewords. In a synchronous multi-user environment,two codewords assigned to a single user have distance 2k,while two codewords assigned to different users have distanceat least 2k-2. Such an assignment of codewords tousers is called a Kirkman signal set when the number of usersaccommodated is the maximum. In this paper, the existence ofKirkman signal sets with k=3 and mas large as possible is settled for all values of .     

Point Code Minimum Steiner Triple Systems

The point code of a Steiner triple system uniquely determines the system when the number of vectors whose weight equals the replication number agrees with the number of points. The existence of a Steiner triple system with this minimum point code property is established for all v 1,3 (mod 6) with v 15.     

Some new perfect Steiner triple systems

In a Steiner triple system STS(v) = (V, B), for each pair {a, b} ⊂ V, the cycle graph G_a,b can be defined as follows. The vertices of G_a,b are V \ {a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25, and 33. We construct perfect STS (v) for v = 79, 139, 367, 811, 1531, 25771, 50923, 61339, and 69991. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 327–330, 1999