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.     

Existence results for KS3(v; 2, 4)s

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) each 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 nonempty cells of the array is a (v, k, λ)-BIBD. The existence question for KS₂(v; μ, λ) has been completely selttled. We are interested in the next case k = 3. The case k = 3 and μ = λ = 1 appears to be quite difficult, although some existence results are available. For λ > 1 and μ ⩾ 1, the problem is more tractable. In this paper, we prove the existence of KS₃(v; 2, 4) for v ≡ 3 (mod 12), v ≡ 6 (mod 60) and v ≡ 9 (mod 96).     

Existence of resolvable optimal strong partially balanced designs

We shall refer to a strong partially balanced design SPBD(v,b,k;λ,0) whose b is the maximum number of blocks in all SPBD(v,b,k;λ,0), as an optimal strong partially balanced design, briefly OSPBD(v,k,λ). Resolvable strong partially balanced design was first formulated by Wang, Safavi-Naini and Pei [Combinatorial characterization of l-optimal authentication codes with arbitration, J. Combin. Math. Combin. Comput. 37 (2001) 205-224] in investigation of l-optimal authentication codes. This article investigates the existence of resolvable optimal strong partially balanced design ROSPBD(v,3,1). We show that there exists an ROSPBD(v,3,1) for any v?3 except v=6,12.     

Super-simple, Pan-orientable, and Pan-decomposable BIBDs with Block Size 4 and Related Structures

Let v, k, and μ be positive integers. A tournament T of order k, briefly k-tournament, is a directed graph on k vertices in which there is exactly one directed edge between any two vertices. A (v, k, λ = 2μ)-BIBD is called T-orientable if for each of its blocks B, it is possible to replace B by a copy of T on the set B so that every ordered pair of distinct points appears in exactly μ k-tournaments. A (v, k, λ = 2μ)-BIBD is called pan-orientable if it is T-orientable for every possible k-tournament T. In this paper, we continue the earlier investigations and complete the spectrum for (v, 4, λ = 2μ)-BIBDs which possess both the pan-orientable property and the pan-decomposable property first introduced by Granville et al. (Graphs Comb 5:57–61, 1989). For all μ, we are able to show that the necessary existence conditions are sufficient. When λ = 2 and v > 4, our designs are super-simple, that is they have no two blocks with more than two common points. One new corollary to this result is that there exists a (v, 4, 2)-BIBD which is both super-simple and directable for all v ≡ 1, 4 (mod 6), v > 4. Finally, we investigate the existence of pan-orientable, pan-decomposable (v, 4, λ = 2μ)-BIBDs with a pan-orientable, pan-decomposable (w, 4, λ = 2μ)-BIBD as a subdesign; here we obtain complete results for λ = 2, 4, but there remain several open cases for λ = 6 (mostly for v < 4w), and the case λ = 12 still has to be investigated.     

The Existence of Kirkman Squares—Doubly Resolvable (v,3,1)-BIBDs

A Kirkman square with index , latinicity , block size k, and v points, KS _k (v;,), is a t×t array (t=(v–1)/(k–1)) defined on a v-set V such that (1) every point of V is contained in precisely 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,)-BIBD. For =1, the existence of a KS _k (v; , ) is equivalent to the existence of a doubly resolvable (v, k, )-BIBD. The spectrum of KS ₂ (v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with =1. We show that there exist KS ₃ (v; 1, 1) for , v=3 and v27 with at most 23 possible exceptions for v.     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

On three-designs of small order

For positive integers t?k?v and λ we define a t-design, denoted B_i[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_i:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_i|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs B_t[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed.     

Resolvable balanced bipartite designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets, B¹ and B², where ∣Bⁱ∣ = k_i, i = 1 or 2. Two elements are said to be linked in B if and only if they belong to different subsets of B. A balanced bipartite design, BBD(v, k₁, k₂, λ), is an arrangement of v elements into b blocks, each containing k elements such that each element occurs in exactly r blocks and any two distinct elements are linked in exactly λ blocks. A resolvable balanced bipartite design, RBBD(v, k₁, k₂, λ), is a BBD(v, k₁, k₂, λ), the b blocks of which can be divided into r sets which are called complete replications, such that each complete replication contains all the v elements of the design.Necessary conditions for the existence of RBBD(v, 1, k₂, λ) and RBBD(v, n, n, λ) are obtained and it is shown that some of the conditions are also sufficient. In particular, necessary and sufficient conditions for the existence of RBBD(v, 1, k₂, λ), where k₂ is odd or equal to two, and of RBBD(v, n, n, λ), where n is even and 2n ? 1 is a prime power, are given.     

A. translation of J. X. Lu's “an existence theory for resolvable balanced incomplete block designs”

In 1984, J. X. Lu proved the following statement. Given any k and λ, there exists a constant c(k, λ) such that an RB[v,k,λ] exists for all v > c(k,λ) satisfying the usual necessary conditions. Lu's paper was written in Chinese and, unfortunately, its accessibility is limited. We have translated Lu's paper into English and have also given a new interpretation of his constructions for resolvable block designs. © 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.     

An existence theory for pairwise balanced designs,III: Proof of the existence conjectures

Given positive integers k and λ, balanced incomplete block designs on v points with block size k and index λ exist for all sufficiently large integers v satisfying the congruences λ(v ? 1) ≡ 0 (mod k ? 1) and λv(v ? 1) ≡ 0 (mod k(k ? 1)). Analogous results hold for pairwise balanced designs where the block sizes come from a given set K of positive integers. We also observe that the number of nonisomorphic designs on v points with given block size k > 2 and index λ tends to infinity as v increases (subject to the above congruences).     

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.     

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.     

Resolvable balanced incomplete block designs with subdesigns of block size 4

In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly (v,4,1) RBIBDs. The problem we investigate is the existence of (v,4,1) RBIBDs containing a (w,4,1) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB(v,w). The necessary conditions for the existence of an IRB(v,w) are that v?4w and . We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever and whenever w?1852.We also give some results on pairwise balanced designs on v points containing (at least one) block of size w, i.e., a (v,{K,w^*},1)-PBD.If the list of permitted block sizes, K₅, contains all integers of size 5 or more, and v,w∈K₅, then a necessary condition on this PBD is v?4w+1. We show this condition is not sufficient for any w?5 and give the complete spectrum (in v) for 5?w?8, as well as showing the condition v?5w is sufficient with some definite exceptions for w=5 and 6, and some possible exceptions when w=15, namely 77?v?79. The existence of this PBD implies the existence of an IRB(12v+4,12w+4).If the list of permitted block sizes, K₁₍₄₎, contains all integers , and v,w∈K₁₍₄₎, then a necessary condition on this PBD is v?4w+1. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w?1037. The existence of this PBD implies the existence of an IRB(3v+1,3w+1).     

Small Uniformly Resolvable Designs for Block Sizes 3 and 4

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k‐pc and for a given k the number of k‐pcs is denoted r_k. In this paper, we consider the case of block sizes 3 and 4 (both existent). We use v to denote the number of points, in this case the necessary conditions imply that v ≡ 0 (mod 12). We prove that all admissible URDs with v < 200 points exist, with the possible exceptions of 13 values of r₄ over all permissible v. We obtain a URD({3, 4}; 276) with r₄ = 9 by direct construction use it to and complete the construction of all URD({3, 4}; v) with r₄ = 9. We prove that all admissible URDs for v ≡ 36 (mod 144), v ≡ 0 (mod 60), v ≡ 36 (mod 108), and v ≡ 24 (mod 48) exist, with a few possible exceptions. Recently, the existence of URDs for all admissible parameter sets with v ≡ 0 (mod 48) was settled, this together with the latter result gives the existence all admissible URDs for v ≡ 0 (mod 24), with a few possible exceptions.     

Super-simple balanced incomplete block designs with block size 4 and index 5

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple designs are also useful in other constructions, such as superimposed codes and perfect hash families etc. The existence of super-simple (v,4,λ)-BIBDs have been determined for λ=2,3,4 and 6. When λ=5, the necessary conditions of such a design are that and v≥13. In this paper, we show that there exists a super-simple (v,4,5)-BIBD for each and v≥13.     

On the construction of handcuffed designs

A handcuffed design with parameters ν, k, λ consists of a set of ordered k-subsets of a v-set, called handcuffed blocks; in a block (a₁, a₂, a_k) each element is assumed to be “handcuffed” to its neighbors. A block, therefore, contains k ? 1 handcuffed pairs, the pairs being considered unordered. Each element of the v-set appears in exactly r blocks, and each pair of distinct elements of the v-set is handcuffed in exactly A blocks of the design.These designs have been studied recently by Hung and Mendelsohn [1], who construct a number of families of such designs by recursive methods. In this paper we show how difference methods can be applied to the construction of handcuffed designs. The methods are powerful, and a number of families of designs are constructed. A main new result is the determination of necessary and sufficient conditions for the existence of handcuffed designs for all parameter sets in which v is an odd prime power.     

Resolvability of infinite designs

In this paper we examine the resolvability of infinite designs. We show that in stark contrast to the finite case, resolvability for infinite designs is fairly commonplace. We prove that every t  -(v,k,Λ)$(v,k,\Lambda )$ design with t finite, v   infinite and k,λ<v$k,\lambda <v$ is resolvable and, in fact, has α   orthogonal resolutions for each α<v$\alpha <v$. We also show that, while a t  -(v,k,Λ)$(v,k,\Lambda )$ design with t and λ finite, v   infinite and k=v$k=v$ may or may not have a resolution, any resolution of such a design must have v parallel classes containing v   blocks and at most λ−1$\lambda -1$ parallel classes containing fewer than v   blocks. Further, a resolution into parallel classes of any specified sizes obeying these conditions is realisable in some design. When k<v$k<v$ and λ=v$\lambda =v$ and when k=v$k=v$ and λ is infinite, we give various examples of resolvable and non-resolvable t  -(v,k,Λ)$(v,k,\Lambda )$ designs.