Pitch tournament designs and other BIBDs—existence results for the case ν = 8n

A pitch tournament is a resolvable or near resolvable(ν,8,7) BIBD that satisfies certain criteria in addition to theusual condition that ν ≡ 0 or 1 (mod 8). Here we establish that for the case ν = 8n the necessary condition forpitch tournaments is sufficient for all n > 1615, with at most 187 smaller exceptions. This complements our earlier study of the ν = 8n + 1 case, where we established sufficiency for all n > 224, with at most 28 smaller exceptions. The four missing cases for (ν,8,7) BIBDs are provided, namely ν∈{48,56,96,448}, thereby establishing that the necessary existence conditions are sufficient without exception. Some constructions for resolvable designs are also provided, reducing the existence question for (ν,8,7) RBIBDs to 21 possible exceptions. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 334–356, 2001     

Some new bibds with k = 6 and λ = 1

(v, 6,1) BIBDs are given for several new values of v. This reduces to 55 the number of v values for which existence of a (v,6,1) BIBD is in doubt. A new resolvable (565,5,1) BIBD is also given. © 1995 John Wiley & Sons, Inc.     

α‐Resolvable group divisible designs with block size three

A group divisible design GD(k,λ,t;tu) is α‐resolvable if its blocks can be partitioned into classes such that each point of the design occurs in precisely α blocks in each class. The necessary conditions for the existence of such a design are λt(u ? 1) = r(k ? 1), bk = rtu, k|αtu and α|r. It is shown in this paper that these conditions are also sufficient when k = 3, with some definite exceptions. © 2004 Wiley Periodicals, Inc.     

1‐rotationally resolvable 4‐cycle systems of 2Kv

In this article, it is shown that there exists a 1‐rotationally resolvable 4‐cycle system of 2K_υ if and only if υ ≡ 0 (mod 4). To prove that, some special sequences of integers are utilized. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 116–125, 2002; DOI 10.1002/jcd.10006     

On the existence of resolvable K4 − e designs

In this article, we settle a problem which originated in 4 regarding the existence of resolvable (K₄ ? e)‐design. We solve the problem with two possible exceptions. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 502–510, 2007     

New families of non‐embeddable quasi‐derived designs

The main result in this article is a method of constructing a non‐embeddable quasi‐derived design from a quasi‐derived design and an α‐resolvable design. This method is a generalization of techniques used by van Lint and Tonchev in 14 , 15 and Kageyama and Miao in 8 . As applications, we construct several new families of non‐embeddable quasi‐derived designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 263–275, 2008     

Pairwise balanced designs whose block size set contains seven and thirteen

In this paper, we investigate the PBD‐closure of sets K with {7,13} ? K ? {7,13,19,25,31,37,43}. In particular, we show that ν ≡ 1 mod 6, ν ≥ 98689 implies ν ? B({7,13}). As an intermediate result, many new 13‐GDDs of type 13^q and resolvable BIBD with block size 6 or 12 are also constructed. Furthermore, we show some elements to be not essential in a Wilson basis for the PBD‐closed set {ν: ν ≡ 1 mod 6, ν ≥ 7}. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 283–314, 2007     

Some new BIBDS with λ = 1 and 6 ≤ k ≤ 10

This article is in two main parts. The first gives some (q,k, 1) difference families with q a prime power and 7 ≤ k ≤ 9; it also gives some GD(k, 1, k,kq)s which are extendable to resolvable (kq,k, 1) BIBDs for k E {6,8,10} and q a prime power equal to 1 mod 2(k − 1). The second uses some of these plus several recursive constructions to obtain some new (v,k,, 1) BIBDs with 7 ≤ k ≤ 9 and some new (v,8,1) resolvable BIBDs. © 1996 John Wiley & Sons, Inc.     

Resolvable BIBDs with block size 8 and index 7

The necessary conditions for the existence of a resolvable BIBD RB(k,λ; v) are λ(v ? 1) = 0(mod k ? 1) and v = 0(mod k). In this article, it is proved that these conditions are also sufficient for k = 8 and λ = 7, with at most 36 possible exceptions. © 1994 John Wiley & Sons, Inc.     

Enumeration of resolvable 2‐(10, 5, 16) and 3‐(10, 5, 6) designs

A backtracking over parallel classes with a partial isomorph rejection (PIR) is carried out to enumerate the resolvable 2‐(10,5,16) designs. Computational results show that the inclusion of PIR reduce substantially the CPU time for the enumeration of all designs. We prove first some results, which enable us to restrict the search space. Since every resolvable 2‐(10,5,16) design is also a resolvable 3‐(10,5,6) design and vice versa, the latter designs are also enumerated. There are 27, 121, 734 such designs with automorphism groups whose order range from 1 to 1,440. From these, designs 2,006,690 are simple. © 2004 Wiley Periodicals, Inc.     

Resolvable even‐cycle systems with a 1‐rotational automorphism

In this article, necessary and sufficient conditions for the existence of a 1‐rotationally resolvable even‐cycle system of λK_v are given, which are eventually for the existence of a resolvable even‐cycle system of λK_v. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 394–407, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10058     

Near vector spaces over GF(q) and (v,q+1,1)-BIBD''s

The usual construction of (v,q+1,1)−BIBD's from vector spaces over GF(q) is generalized to the class of near vector spaces over GF(q). It is shown that every (v,q+1,1)−BIBD can be constructed from a near vector space over GF(q). Some corollaries are: Given a (v₁,q+1,1)−BIBD P₁,B₁ and a (v₂,q+1,1)−BIBD P₂,B₂, there is a ((q−1)v₁v₂+v₁+v₂,q+1,1)−BIBD P₃,B₃ containing P₁,B₁ and P₂,B₂ as disjoint subdesigns. If there is a (v,q+1,1)−BIBD then there is a ((q−1)v+1,q,1)−BIBD. Every finite partial (v,q,1)−BIBD can be embedded in a finite (v′,q+1,1)−BIBD.     

A construction of resolvable nested 3‐designs

A method of constructing resolvable nested 3‐designs from an affine resolvable 3‐design is proposed with one example. © 2004 Wiley Periodicals, Inc.     

Orthogonal resolutions in odd balanced tournament designs

An odd balanced tournament design,OBTD(n), is ann × 2n + 1 array of pairs defined on a (2n + 1)-setV such that (1) every row of the array contains each element ofV twice, (2) every column of the array contains 2n distinct elements ofV, and (3) the pairs of the array form a (2n + 1, 2, 1)-BIBD. In this paper, we investigate the spectrum of odd balanced tournament designs with orthogonal resolutions. These designs can be used to construct doubly near resolvable (v, 3, 2)-BIBDs.     

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ‐BIBDs are (mod ) and . In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ‐RBIBD, namely (mod ) and , are sufficient. A few new pairwise balanced designs are also given.     

Cyclically near‐resolvable 4‐cycle systems of 2Kv

Fu and Mishima [J. Combin. Des. 10 (2002), pp. 116–125] have utilized the extended Skolem sequence to prove that there exists a 1‐rotationally resolvable $4$‐cycle system of $2K_v$ if and only if $v\equiv 0$ (mod $4$). In this paper, the existence of a cyclically near‐resolvable $4$‐cycle system is discussed, and it is shown that there exists a cyclically near‐resolvable $4$‐cycle system of $2K_v$ if and only if $v\equiv 1$ (mod $4$).     

Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

Block designs with large holes and α -resolvable BIBDs

In a (v, k, λ: w) incomplete block design (IBD) (or PBD [v, {k, w^*}. λ]), the relation v ≥ (k ? 1)w + 1 must hold. In the case of equality, the IBD is referred to as a block design with a large hole, and the existence of such a configuration is equivalent to the existence of a λ-resolvable BIBD(v ? w, k ? 1, λ). The existence of such configurations is investigated for the case of k = 5. Necessary and sufficient conditions are given for all v and λ ? 2 (mod 4), and for λ ≡ 2 mod 4 with 11 possible exceptions for v. © 1993 John Wiley & Sons, Inc.     

Super‐simple holey Steiner pentagon systems and related designs

A Steiner pentagon system of order v (SPS(v)) is said to be super‐simple if its underlying (v, 5, 2)‐BIBD is super‐simple; that is, any two blocks of the BIBD intersect in at most two points. It is well known that the existence of a holey Steiner pentagon system (HSPS) of type T implies the existence of a (5, 2)‐GDD of type T. We shall call an HSPS of type T super‐simple if its underlying (5, 2)‐GDD of type T is super‐simple; that is, any two blocks of the GDD intersect in at most two points. The existence of HSPSs of uniform type hⁿ has previously been investigated by the authors and others. In this article, we focus our attention on the existence of super‐simple HSPSs of uniform type hⁿ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 301–328, 2008