Existence of directed GDDs with block size five

In this article, we construct directed group divisible designs (DGDDs) with block size five, group-type hⁿ, and index unity. The necessary conditions for the existence of such a DGDD are n ≥ 5, (n − 1)h ≡ 0 (mod 2) and n(n − 1)h² ≡ 0 (mod 10). It is shown that these necessary conditions are also sufficient, except possibly for n = 15 where h ≡ 1 or 5 (mod 6) and h ≢ 0 (mod 5), or (n, h) = (15, 9). © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 389–402, 1998     

Pair covering designs with block size 5

In this article we look at pair covering designs with a block size of 5 and . The number of blocks in a minimum covering design is known as the covering number C(v,5,2). For v?24, these values are known, and all but v=8 exceed the Schönheim bound, L(v,5,2)=⌈v/5⌈(v-1)/4⌉⌉. However, for all v?28 with , it seems probable that C(v,5,2)=L(v,5,2). We establish this for all but 17 possible exceptional values lying in the range 40?v?280.     

Kirkman packing and covering designs with spanned holes of size 2

A Kirkman holey packing (resp. covering) design, denoted by KHPD(g^u) (resp. KHCD(g^u)), is a resolvable (gu, 3, 1) packing (resp. covering) design of pairs with u disjoint holes of size g, which has the maximum (resp. minimum) possible number of parallel classes. Each parallel class contains one block of size δ, while other blocks have size 3. Here δ is equal to 2, 3, and 4 when gu ≡ 2, 3, and 4 (mod 3) in turn. In this paper, the existence problem of a KHPD(2^u) and a KHCD(2^u) is solved with one possible exception of a KHPD(2⁸). © 2004 Wiley Periodicals, Inc.     

Group divisible designs with block size 4 where the group sizes are congruent to 2 mod 3

In this paper, we construct a number of 4-GDDs where the group sizes are all congruent to 2 (mod 3). We also show that 4-GDDs of type $2^t{8}^s$ exist for all but a finite number of feasible values of s and t. The largest unknown case has type $2^4{8}^{18}$ and has 152 points. A number of 4-GDDs with at most 50 points are also constructed. These include one of type $4^8{1}^1{10}^1$, the last feasible type of the form $4^s{1}^t{n}^1$ with at most 50 points for which no 4-GDD was known.     

Incomplete group divisible designs with block size four and general index

In this paper, we investigate the existence of incomplete group divisible designs (IGDDs) with block size four, group-type (g, h) u and general index λ. The necessary conditions for the existence of such a design are that u ≥ 4, g ≥ 3h, λg(u 1) ≡ 0 (mod 3), λ(g h)(u 1) ≡ 0 (mod 3), and λu(u 1)(g 2 h 2 ) ≡ 0 (mod 12). These necessary conditions are shown to be sufficient for all λ≥ 2. The known existence result for λ = 1 is also improved.     

Resolvable modified group divisible designs with block size three

A resolvable modified group divisible design (RMGDD) is an MGDD whose blocks can be partitioned into parallel classes. In this article, we investigate the existence of RMGDDs with block size three and show that the necessary conditions are also sufficient with two exceptions. © 2005 Wiley Periodicals, Inc. J Combin Designs 15: 2–14, 2007     

Group divisible designs with block size four and two groups

We give some constructions of new infinite families of group divisible designs, GDD(n,2,4;λ₁,λ₂), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3?n?8. For n=10 there is one missing critical design. If λ₁>λ₂, then the necessary conditions are sufficient for . For each of n=10,15,16,17,18,19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12.     

Super-simple group divisible designs with block size 4 and index 5

In this paper, we investigate the existence of a super-simple (4, 5)-GDD of type g^u and show that such a design exists if and only if u≥4, g(u−2)≥10, and .     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

区组大小为 4 的二重超单可分解的可分组设计

自 1992 年 Gronau 和 Mullin 提出超单设计的概念以来, 很多研究者参与了超单设计的研究. 超单设计在编码等方面也有广泛的应用. 超单可分组设计是超单设计的重要组成部分. 本文我们主要研究区组大小为4 的二重超单可分解的可分组设计, 并基本解决了此类设计的存在性问题.     

Existence of generalized Bhaskar Rao designs with block size 3

There are well-known necessary conditions for the existence of a generalized Bhaskar Rao design over a group G, with block size k=3. The recently proved Hall-Paige conjecture shows that these are sufficient when v=3 and λ=|G|. We prove these conditions are sufficient in general when v=3, and also when |G| is small, or when G is dicyclic. We summarize known results supporting the conjecture that these necessary conditions are always sufficient when k=3.     

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     

On small packing and covering designs with block size 4

A packing (respectively covering) design of order v, block size k, and index λ is a collection of k-element subsets, called blocks, of a v-set, V, such that every 2-subset of V occurs in at most (at least) λ blocks. The packing (covering) problem is to determine the maximum (minimum) number of blocks in a packing (covering) design. Motivated by the recent work of Assaf [1] [2], we solve the outstanding packing and covering problems with block size 4.     

Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032     

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.     

Uniformly resolvable designs with index one and block sizes three and four — with three or five parallel classes of block size four

Each parallel class of a uniformly resolvable design (URD) contains blocks of only one block size. A URD with v points and with block sizes three and four means that at least one parallel class has block size three and at least one has block size four. Danziger [P. Danziger, Uniform restricted resolvable designs with r=3, ARS Combin. 46 (1997) 161-176] proved that for all there exist URDs with index one, some parallel classes of block size three, and exactly three parallel classes with block size four, except when v=12 and except possibly when . We extend Danziger’s work by showing that there exists a URD with index one, some parallel classes with block size three, and exactly three parallel classes with block size four if, and only if, , v≠12. We also prove that there exists a URD with index one, some parallel classes of block size three, and exactly five parallel classes with block size four if, and only if, , v≠12. New labeled URDs, which give new URDs as ingredient designs for recursive constructions, are the key in the proofs. Some ingredient URDs are also constructed with difference families.     

Embeddings of resolvable group divisible designs with block size 3

It is proved in this paper that an RGD(3, g;v) can be embedded in an RGD(3, g;u) if and only if , , , v ≥ 3g, u ≥ 3v, and (g,v) ≠ (2,6),(2,12),(6,18).     

On constructions for two dimensional balanced sampling plan excluding contiguous units with block size four

Balanced sampling plans excluding contiguous units (BSEC) were first introduced in 1988 by Hedayat, Rao and Stufken [A.S. Hedayat, C.R. Rao, J. Stufken, Sampling plans excluding contiguous units, J. Statist. Plann. Inference 19 (1988) 159-170]. These designs can be used for survey sampling when the contiguous units provide similar information. In this paper, we show some recursive constructions for two dimensional BSECs with block size four, and give the existence of some infinite classes.