On 6-sparse Steiner triple systems

We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also made concerning existing construction methods for perfect Steiner triple systems, and we give a further example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a uniform Steiner triple system of order 180,907.     

Nonexistence of sparse triple systems over abelian groups and involutions

In 1973 Paul Erdős conjectured that there is an integer v ₀(r) such that, for every v>v ₀(r) and v≡1,3 (mod 6), there exists a Steiner triple system of order v, containing no i blocks on i+2 points for every 1<i≤r. Such an STS is said to be r-sparse. In this paper we consider relations of automorphisms of an STS to its sparseness. We show that for every r≥13 there exists no point-transitive r-sparse STS over an abelian group. This bound and the classification of transitive groups give further nonexistence results on block-transitive, flag-transitive, 2-transitive, and 2-homogeneous STSs with high sparseness. We also give stronger bounds on the sparseness of STSs having some particular automorphisms with small groups. As a corollary of these results, it is shown that various well-known automorphisms, such as cyclic, 1-rotational over arbitrary groups, and involutions, prevent an STS from being high-sparse.      

Existence of non‐resolvable Steiner triple systems

We consider two well‐known constructions for Steiner triple systems. The first construction is recursive and uses an STS(v) to produce a non‐resolvable STS(2v + 1), for v ≡ 1 (mod 6). The other construction is the Wilson construction that we specify to give a non‐resolvable STS(v), for v ≡ 3 (mod 6), v > 9. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 16–24, 2005.     

Large sets with holes

We study large sets of disjoint Steiner triple systems “with holes”. The purpose of these structures is to extend the use of large sets of disjoint Steiner triple systems in the construction of various other large set type structures to values of v for which no Steiner triple system of order v exists, i.e., v ≡ 0, 2, 4, or 5 (mod 6). We give constructions for all of these congruence classes. We describe several applications, including applications to large sets of disjoint group divisible designs and to large sets of disjoint separable ordered designs. © 1993 John Wiley & Sons, Inc.     

Primitive symmetric designs with up to 2500 points

By this article we conclude the construction of all primitive ( v, k,λ ) symmetric designs with v < 2500 , up to a few unsolved cases. Complementary to the designs with prime power number of points published previously, here we give 55 primitive symmetric designs with v ≠ p ^m , p prime and m positive integer, together with the analysis of their full automorphism groups. The research involves programming and wide‐range computations. We make use of the software package GAP and the library of primitive groups which it contains. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:463‐474, 2011     

The resolution of the anti‐Pasch conjecture

We show that an anti‐Pasch Steiner triple system of order v exists for v ≡ 1 or 3 (mod 6), apart from v = 7 and 13. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 300–309, 2000     

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.     

Hanani triple systems

Hanani triple systems onv≡1 (mod 6) elements are Steiner triple systems having (v−1)/2 pairwise disjoint almost parallel classes (sets of pairwise disjoint triples that spanv−1 elements), and the remaining triples form a partial parallel class. Hanani triple systems are one natural analogue of the Kirkman triple systems onv≡3 (mod 6) elements, which form the solution of the celebrated Kirkman schoolgirl problem. We prove that a Hanani triple system exists for allv≡1 (mod 6) except forv ∈ {7, 13}.     

On a More General Characterisation of Steiner Systems

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 be a [k,2(k−t+1)]-sparse geometry of cardinality n, with k \> t \> 1$$" align="middle" border="0"> . If , 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 and link every block with the code word 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).     

Pasch trades with a negative block

A Steiner triple system of order v, STS(v), may be called equivalent to another STS(v) if one can be converted to the other by a sequence of three simple operations involving Pasch trades with a single negative block. It is conjectured that any two STS(v)s on the same base set are equivalent in this sense. We prove that the equivalence class containing a given system S on a base set V contains all the systems that can be obtained from S by any sequence of well over one hundred distinct trades, and that this equivalence class contains all isomorphic copies of S on V. We also show that there are trades which cannot be effected by means of Pasch trades with a single negative block.     

Covering arrays of strength 3 and 4 from holey difference matrices

A covering array CA(N; t, k, v) is an N × k array with entries from a set X of v symbols such that every N × t sub-array contains all t-tuples over X at least once, where t is the strength of the array. The minimum size N for which a CA(N; t, k, v) exists is called the covering array number and denoted by CAN(t, k, v). Covering arrays are used in experiments to screen for interactions among t-subsets of k components. One of the main problems on covering arrays is to construct a CA(N; t, k, v) for given parameters (t, k, v) so that N is as small as possible. In this paper, we present some constructions of covering arrays of strengths 3 and 4 via holey difference matrices with prescribed properties. As a consequence, some of known bounds on covering array number are improved. In particular, it is proved that (1) CAN(3, 5, 2v) ≤ 2v ²(4v + 1) for any odd positive integer v with gcd(v, 9) ≠ 3; (2) CAN(3, 6, 6p) ≤ 216p ³ + 42p ² for any prime p > 5; and (3) CAN(4, 6, 2p) ≤ 16p ⁴ + 5p ³ for any prime p ≡ 1 (mod 4) greater than 5.     

The permutahedron ofN-sparse posets

The permutahedron of a poset is the convex hull of all incidence vectors of linear extensions. For the case ofN-sparse posets in which any five elements induce at most oneN we give a characterization of the permutahedron in terms of linear inequalities. This yields an LP-solution for minimizing the weighted mean completion time for jobs with unit processing times andN-sparse precedence constraints. We close with an extension of our approach to arbitrary processing times.     

Distance and fractional isomorphism in Steiner triple systems

In [8], Quattrochi and Rinaldi introduced the idea ofn ⁻¹-isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integerN, there existsv ₀(N) such that for all admissiblev≥v ₀(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn ⁻¹-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2⁻¹-isomorphic. Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2⁻¹-isomorphic and 3⁻¹-isomorphic pairs of STS(15)s.     

Primitive symmetric designs with prime power number of points

In this paper we either prove the non‐existence or give explicit construction of primitive symmetric (v, k, λ) designs with v=p^m<2500, p prime and m>1. The method of design construction is based on an automorphism group action; non‐existence results additionally include the theory of difference sets, multiplier theorems in particular. The research involves programming and wide‐range computations. We make use of software package GAP and the library of primitive groups which it contains. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 141–154, 2010     

A completion of the spectrum for large sets of disjoint transitive triple systems

In this paper, we give a recursive construction from an LTTS(v + 2) to an LTTS(16v + 2) for v 3. Furthermore, the existence of LTTS(2ⁿ + 2) is proved. Thereby, we completely solve the existence problem of LTTS)(v) (large set of pairwise disjoint transitive triple systems of order v).     

Another construction for large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection , where every is a KTS(v) and all form a partition of all triples on X. In this article, we give a new construction for LKTS(6v + 3) via OLKTS(2v + 1) with a special property and obtain new results for LKTS, that is there exists an LKTS(3v) for , where p, q ≥ 0, r _i , s _j ≥ 1, q _i is a prime power and mod 12.      

A new existence proof for large sets of disjoint Steiner triple systems

A Steiner triple system of order v (briefly STS(v)) consists of a v-element set X and a collection of 3-element subsets of X, called blocks, such that every pair of distinct points in X is contained in a unique block. A large set of disjoint STS(v) (briefly LSTS(v)) is a partition of all 3-subsets (triples) of X into v-2 STS(v). In 1983–1984, Lu Jiaxi first proved that there exists an LSTS(v) for any v≡1 or with six possible exceptions and a definite exception v=7. In 1989, Teirlinck solved the existence of LSTS(v) for the remaining six orders. Since their proof is very complicated, it is much desired to find a simple proof. For this purpose, we give a new proof which is mainly based on the 3-wise balanced designs and partitionable candelabra systems.     

On the holomorphic extension of CR functions from non-generic CR submanifolds of ? n

We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold N of ℂ ⁿ to complex transversal wedges with edges containing N. We show that given any v∈ℂ ⁿ ∖(T _p N+iT _p N), there exists a wedge of direction v whose edge contains a neighborhood of p in N, such that any continuous CR function defined locally near p extends holomorphically to that wedge.     

Partitions of sets of two-fold triple systems,and their relation to some strongly regular graphs

LetX be a set ofv + 1 elements, wherev 0 or 1 (mod 3). If two copies of the collection of triples chosen fromX can be partitioned intov + 1 mutually disjoint two-fold triple systems, each based on a differentv-subset ofX, we say they form an overlarge set of two-fold triple systems, denoted byOS(TTS(v)). In this paper, we give the first construction of anOS(TTS(10)). We then investigate the properties of the uniqueOS(TTS(6)) and obtain:

Some progress on the existence of 1-rotational Steiner triple systems

A Steiner triple system of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point. The spectrum of values of v for which there exists a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57–66, 1981), Buratti (J Combin Des 9:215–226, 2001) and Mishima (Discrete Math 308:2617–2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-rotational STS(v) under an arbitrary group has not been completely determined yet. This paper is a considerable step forward to the solution of this problem. In fact, we leave as uncertain cases only those for which we have v =  (p ³−p)n +  1 ≡ 1 (mod 96) with p a prime, n \not o 0{n \not\equiv 0} (mod 4), and the odd part of (p ³ − p)n that is square-free and without prime factors congruent to 1 (mod 6).