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.     

The existence of doubly near resolvable (v,3,2)-BIBDs

The existence of doubly near resolvable (v,2,1)-BIBDs was established by Mullin and Wallis in 1975. In this article, we determine the spectrum of a second class of doubly near resolvable balanced incomplete block designs. We prove the existence of DNR(v,3,2)-BIBDs for v ≡ 1 (mod 3), v ≥ 10 and v ? {34,70,85,88,115,124,133,142}. The main construction is a frame construction, and similar constructions can be used to prove the existence of doubly resolvable (v,3,2)-BIBDs and a class of Kirkman squares with block size 3, KS₃(v,2,4). © 1994 John Wiley & Sons, Inc.     

Large sets of resolvable idempotent Latin squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the v(v−1) off-diagonal cells can be resolved into v−1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of v−2 RILS(v)s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.     

Existence of resolvable H-designs with group sizes 2, 3, 4 and 6

In 1987, Hartman showed that the necessary condition v ≡ 4 or 8 (mod 12) for the existence of a resolvable SQS(v) is also sufficient for all values of v, with 23 possible exceptions. These last 23 undecided orders were removed by Ji and Zhu in 2005 by introducing the concept of resolvable H-designs. In this paper, we first develop a simple but powerful construction for resolvable H-designs, i.e., a construction of an RH(g ²ⁿ ) from an RH((2g) ⁿ ), which we call group halving construction. Based on this construction, we provide an alternative existence proof for resolvable SQS(v)s by investigating the existence problem of resolvable H-designs with group size 2. We show that the necessary conditions for the existence of an RH(2 ⁿ ), namely, n ≡ 2 or 4 (mod 6) and n ≥ 4 are also sufficient. Meanwhile, we provide an alternative existence proof for resolvable H-designs with group size 6. These results are obtained by first establishing an existence result for resolvable H-designs with group size 4, that is, the necessary conditions n ≡ 1 or 2 (mod 3) and n ≥ 4 for the existence of an RH(4 ⁿ ) are also sufficient for all values of n except possibly n ∈ {73, 149}. As a consequence, the general existence problem of an RH(g ⁿ ) is solved leaving mainly the case of g ≡ 0 (mod 12) open. Finally, we show that the necessary conditions for the existence of a resolvable G-design of type g ⁿ are also sufficient.     

Super-Simple Resolvable Balanced Incomplete Block Designs with Block Size 4 and Index 4

The necessary conditions for the existence of a super-simple resolvable balanced incomplete block design on v points with block size k = 4 and index λ = 4, are that v ≥ 16 and v ≡ 4 (mod 12). These conditions are shown to be sufficient.     

Maximal resolvable packings and minimal resolvable coverings of triples by quadruples

Determination of maximal resolvable packing number and minimal resolvable covering number is a fundamental problem in designs theory. In this article, we investigate the existence of maximal resolvable packings of triples by quadruples of order v (MRPQS(v)) and minimal resolvable coverings of triples by quadruples of order v (MRCQS(v)). We show that an MRPQS(v) (MRCQS(v)) with the number of blocks meeting the upper (lower) bound exists if and only if v≡0 (mod 4). As a byproduct, we also show that a uniformly resolvable Steiner system URS(3, {4, 6}, {r₄, r₆}, v) with r₆≤1 exists if and only if v≡0 (mod 4). All of these results are obtained by the approach of establishing a new existence result on RH(6²ⁿ) for all n≥2. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 209–223, 2010     

The existence of resolvable Steiner quadruple systems

A Steiner quadruple system of order v is a set X of cardinality v, and a set Q, of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. A Steiner quadruple system is resolvable if Q can be partitioned into parallel classes (partitions of X). A necessary condition for the existence of a resolvable Steiner quadruple system is that v≡4 or 8 (mod 12). In this paper we show that this condition is also sufficient for all values of v, with 24 possible exceptions.     

General frame constructions for Z-cyclic triplewhist tournaments

Frames are useful in dealing with resolvable designs such as resolvable balanced incomplete block designs and triplewhist tournaments. Z-cyclic triplewhist tournament frames are also useful in the constructions of Z-cyclic triplewhist tournaments. In this paper, the concept of an (h₁,h₂,…,hn;u)-regular Z-cyclic triplewhist tournament frame is defined, and used to establish several quite general recursive constructions for Z-cyclic triplewhist tournaments. As corollaries, we are able to unify many known constructions for Z-cyclic triplewhist tournaments. As an application, some new Z-cyclic triplewhist tournament frames and Z-cyclic triplewhist tournaments are obtained. The known existence results of such designs are then extended.     

Unifying some known infinite families of combinatorial 3-designs

In this paper we present a construction of 3-designs by using a 3-design with resolvability. The basic construction generalizes a well-known construction of simple 3-(v,4,3) designs by Jungnickel and Vanstone (1986). We investigate the conditions under which the designs obtained by the basic construction are simple. Many infinite families of simple 3-designs are presented, which are closely related to some known families by Iwasaki and Meixner (1995), Laue (2004) and van Tran (2000, 2001). On the other hand, the designs obtained by the basic construction possess various properties: A theory of constructing simple cyclic 3-(v,4,3) designs by Köhler (1981) can be readily rebuilt from the context of this paper. Moreover many infinite families of simple resolvable 3-designs are presented in comparison with some known families. We also show that for any prime power q and any odd integer n there exists a resolvable 3-(qn+1,q+1,1) design. As far as the authors know, this is the first and the only known infinite family of resolvable t-(v,k,1) designs with t?3 and k?5. Those resolvable designs can again be used to obtain more infinite families of simple 3-designs through the basic construction.     

Existence results for near resolvable designs

A near resolvable design, NRB(v, k), is a balanced incomplete block design whose block set can be partitioned into v classes such that each class contains every point of the design but one, and each point is missing from exactly one class. The necessary conditions for the existence of near resolvable designs are v ≡ 1 mod k and λ = k ? 1. These necessary conditions have been shown to be sufficient for k ? {2,3,4} and almost always sufficient for k ? {5,6}. We are able to show that there exists an integer n₀(k) so that NRB(v,k) exist for all v > n₀(k) and v ≡ 1 mod k. Using some new direct constructions we show that there are many k for which it is easy to compute an explicit bound on n₀(k). These direct constructions also allow us to build previously unknown NRB(v,5) and NRB(v,6). © 1995 John Wiley & Sons, Inc.     

On steiner spaces

A Steiner space is a Steiner triple system that is not generated by a triangle. We give new constructions of Steiner spaces and solve the existence problem of Steiner spaces of order v for all but 4 values of v.     

Uniformly resolvable designs with index one, block sizes three and five and up to five parallel classes with blocks of size five

Each parallel class of a uniformly resolvable design (URD) contains blocks of only one block size k (denoted k-pc). The number of k-pcs is denoted r_k. The necessary conditions for URDs with v points, index one, blocks of size 3 and 5, and r₃,r₅>0, are . If r_k>1, then v≥k², and r₃=(v−1−4⋅r₅)/2. For r₅=1 these URDs are known as group divisible designs. We prove that these necessary conditions are sufficient for r₅=3 except possibly v=105, and for r₅=2,4,5 with possible exceptions (v=105,165,285,345) New labeled frames and labeled URDs, which give new URDs as ingredient designs for recursive constructions, are the key in the proofs.     

A few more Kirkman squares and doubly near resolvable BIBDs with block size 3

A Kirkman square with index λ, latinicity μ, block size k, and v points, KS_k(v;μ,λ), is a t×t array (t=λ(v-1)/μ(k-1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v,k,λ)-BIBD. In a series of papers, Lamken established the existence of the following designs: KS₃(v;1,2) with at most six possible exceptions [E.R. Lamken, The existence of doubly resolvable (v,3,2)-BIBDs, J. Combin. Theory Ser. A 72 (1995) 50-76], KS₃(v;2,4) with two possible exceptions [E.R. Lamken, The existence of KS₃(v;2,4)s, Discrete Math. 186 (1998) 195-216], and doubly near resolvable (v,3,2)-BIBDs with at most eight possible exceptions [E.R. Lamken, The existence of doubly near resolvable (v,3,2)-BIBDs, J. Combin. Designs 2 (1994) 427-440]. In this paper, we construct designs for all of the open cases and complete the spectrum for these three types of designs. In addition, Colbourn, Lamken, Ling, and Mills established the spectrum of KS₃(v;1,1) in 2002 with 23 possible exceptions. We construct designs for 11 of the 23 open cases.     

Maximum uniformly resolvable designs with block sizes 2 and 4

A central question in design theory dating from Kirkman in 1850 has been the existence of resolvable block designs. In this paper we will concentrate on the case when the block size k=4. The necessary condition for a resolvable design to exist when k=4 is that v≡4mod12; this was proven sufficient in 1972 by Hanani, Ray-Chaudhuri and Wilson [H. Hanani, D.K. Ray-Chaudhuri, R.M. Wilson, On resolvable designs, Discrete Math. 3 (1972) 343-357]. A resolvable pairwise balanced design with each parallel class consisting of blocks which are all of the same size is called a uniformly resolvable design, a URD. The necessary condition for the existence of a URD with block sizes 2 and 4 is that v≡0mod4. Obviously in a URD with blocks of size 2 and 4 one wishes to have the maximum number of resolution classes of blocks of size 4; these designs are called maximum uniformly resolvable designs or MURDs. So the question of the existence of a MURD on v points has been solved for by the result of Hanani, Ray-Chaudhuri and Wilson cited above. In the case this problem has essentially been solved with a handful of exceptions (see [G. Ge, A.C.H. Ling, Asymptotic results on the existence of 4-RGDDs and uniform 5-GDDs, J. Combin. Des. 13 (2005) 222-237]). In this paper we consider the case when and prove that a exists for all u≥2 with the possible exception of u∈{2,7,9,10,11,13,14,17,19,22,31,34,38,43,46,47,82}.     

Super‐simple resolvable balanced incomplete block designs with block size 4 and index 3

The necessary conditions for the existence of a super‐simple resolvable balanced incomplete block design on v points with k = 4 and λ = 3, are that v ≥ 8 and v ≡ 0 mod 4. These conditions are shown to be sufficient except for v = 12. © 2003 Wiley Periodicals, Inc.     

On some constructions of modular hadamard matrices

Some new constructions for modular Hadamard and Hadamard matrices are given. Incidentally, it is shown that the Williamson series of H-matrices can also be constructed by using modular Hadamard matrices and resolvable semi-regular group divisible designs.     

Realizations for direct constructions of resolvable Steiner 2‐designs with block size 5

We consider direct constructions due to R. J. R. Abel and M. Greig, and to M. Buratti, for ({ν},5,1) balanced incomplete block designs. These designs are defined using the prime fields F_p for certain primes p, are 1‐rotational over G ⊕ F_p where G is a group of order 4, and are also resolvable under certain conditions. We introduce specifications to the constructions and, by means of character sum arguments, show that the constructions yield resolvable designs whenever p is sufficiently large. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:207–217, 2000     

Embeddings of resolvable mendelsohn triple systems

In this article it is shown that any resolvable Mendelsohn triple system of order u can be embedded in a resolvable Mendelsohn triple system of order v iff v≥ 3u, except possibly for 71 values of (u,v). © 1993 John Wiley & Sons, Inc.     

Embeddable transversal designs

The embeddability of certain (group) divisible designs in symmetric 2-designs is investigated. These designs are symmetric resolvable transversal designs. It is proved that all such transversal designs with v = 2k are embeddable and some necessary and sufficient conditions for other cases are given.