Families of Hadamard group divisible designs

By a family of designs we mean a set of designs whose parameters can be represented as functions of an auxiliary variable t where the design will exist for infinitely many values of t. The best known family is probably the family of finite projective planes with υ = b = t² + t + 1, r = k = t + 1, and λ = 1. In some instances, notably coding theory, the existence of families is essential to provide the degree of precision required which can well vary from one coding problem to another. A natural vehicle for developing binary codes is the class of Hadamard matrices. Bush (1977) introduced the idea of constructing semi-regular designs using Hadamard matrices whereas the present study is concerned mostly with construction of regular designs using Hadamard matrices. While codes constructed from these designs are not optimal in the usual sense, it is possible that they may still have substantial value since, with different values of λ₁ and λ₂, there are different error correcting capabilities.     

On the enumeration of some D-optimal designs

Two matrices with elements taken from the set {-1,1}$\{-1,1\}$ are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of matrices having maximal determinant. We establish that there are 7 equivalence classes for matrices of order 21 and that there are at least 9884 equivalence classes for matrices of order 26. The latter result is obtained primarily using a switching technique for producing new designs from old.     

Recursive constructions for cyclic block designs

Simple recursive constructions for cyclic block designs are given. These yield many new infinite families of cyclic Steiner 2-designs.     

Generalized Bhaskar Rao designs of block size three

We show that the necessary conditions

$\text{λ≡0 (}\text{mod}\text{|G|)}$

,

$\text{λ(υ?1)≡0 (}\text{mod}\text{2)}$

,

$\text{λυ(υ?1)≡}\text{0 (}\text{mod}\text{6)}\text{for}\text{|G|}\text{odd}\text{,}\text{0 (}\text{mod}\text{24)}\text{for}\text{|G|}\text{even}$

, are sufficient for the existence of a generalized Bhaskar Rao design GBRD(υ,b,r,3,λ;G) for the elementary abelian group G, of each order |G|.     

Linear sufficiency and completeness in the context of estimating the parametric function in the general Gauss–Markov model

In this paper we consider linear sufficiency and linear completeness in the context of estimating the estimable parametric function K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$ under the general Gauss–Markov model {y,Xβ,σ²V}$\{\mathbf{y},\mathbf{X}\mathit{\beta },{\sigma }^2\mathbf{V}\}$. We give new characterizations for linear sufficiency, and define and characterize linear completeness in a case of estimation of K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$. Also, we consider a predictive approach for obtaining the best linear unbiased estimator of K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$, and subsequently, we give the linear analogues of the Rao–Blackwell and Lehmann–Scheffé Theorems in the context of estimating K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$.     

The family of t-designs - part II

This is a continuation of Part I of our previous paper (1980) on conbinatorial aspects of t-designs with t ≥ 2. Along the same developments as in Part I, this paper provides tables, assorted examples of t-designs, and interesting unsolved problems. A brief background and a short review of the literature related to each unsolved problem is also provided. Some tables list parameters of designs whose existence or nonexistence is not established in literature. Notation and numbering of Part I are still used here.     

Minimal 2-coverings of a finite affine space based on GF(2)

Let EG(m, 2) denote the m-dimensional finite Euclidean space (or geometry) based on GF(2), the finite field with elements 0 and 1. Let T be a set of points in this space, then T is said to form a q-covering (where q is an integer satisfying 1?q?m) of EG(m, 2) if and only if T has a nonempty intersection with every (m-q)-flat of EG(m, 2). This problem first arose in the statistical context of factorial search designs where it is known to have very important and wide ranging applications. Evidently, it is also useful to study this from the purely combinatorial point of view. In this paper, certain fundamental studies have been made for the case when q=2. Let N denote the size of the set T. Given N, we study the maximal value of m.     

Families of regular designs based on geometries

Recently Bush and Ostrom (1979) settled most of the open questions with respect to inequivalent solutions of a class of semiregular (SR) designs which can be constructed from nets. This paper is a study of the same nature for two families of regular (R) designs derived from finite projective planes. One family presents no problems, but the other which is a ‘double’ family with two parameters is much more difficult. In fact it is here solved only for designs based on planes of orders 3, 4, 5 and 8. Certain general methods exist which are indicated, but we were unable to resolve even the case 7 using this technique.Basically we show the existence of either inequivalent solutions or show there is but one solution settling a number of open cases. In particular for the case λ₁ = 2, λ₂ = 1 we give new solutions to a number of D(2) designs or group divisible designs with two associate classes which have no repeated blocks in contrast with the published solutions which have this undesirable property for a number of applications.     

Projective (2n,n,λ,1)-designs

A projective (2n,n,λ,1)-design is a set $\text{D}$ of n element subsets (called blocks) of a 2n-element set V having the properties that each element of V is a member of λ blocks and every two blocks have a non-empty intersection. This paper establishes existence and non-existence results for various projective (2n,n,λ,1)-designs and their subdesigns.     

On block designs arising from rank 3 graphs

In this paper a criterion showing when the orbit of a subgraph of a given rank 3 graph forms a block design, is given. As an application several classes of block designs are derived from the triangular graph T(n) and the lattice graph L₂(n).     

Regular group divisible designs and Bhaskar Rao designs with block size three

Some recursive constructions are given for Bhaskar Rao designs. Using examples of these designs found by Shyam J. Singh, Rakesh Vyas and new ones given here we show the necessary conditions λ≡0 (mod 2), λυ(υ?1)≡0 (mod 24) are sufficient for the existence of Bhaskar Rao designs with one association class and block size 3. This result is used with a result of Street and Rodger to obtain regular partially balanced block designs with 2υ treatments, block size 3, λ₁=0, group size 2 and υ groups.     

Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

Four mols of order 10 with a hole of order 2

A transversal design with hole T[6;10]?T[6;2] is constructed from a separable group divisible design GD[5,1,8;48].     

Construction of (M,S)-optimal design for block size 3

(M,S)-optimal designs are constructed for block size three when the number of treatments is of the form 6t + 3.     

Some bounds on balanced block designs

Bounds on the latest root of the C-matrix and the number of blocks for a variance-balanced block design are given. These results contain the well-known results as special cases.     

Symmetric designs of types V and VI

In Butler (1984a) a semi-translation block was defined and a classification given of all symmetric 2-(υ,k,λ) designs with λ>1, which contain more than one such block. In this paper we consider symmetric designs of type V and VI. We show that symmetric designs of type V are also of type VI, and in addition we show that all such designs can be obtained from a P_n,q by a construction which we give. Finally examples of proper symmetric designs of type V which are not of type VI are given.     

Resolvable systems of 8-tuples

Let R be a family of k-element blocks of a v-element set V such that any two elements of V are contained in λ blocks of R and R=R₁∪…∪R_v?1, R_i∩ R_j=? (i≠j) and ?{B_i∈R_j∣i=1,…,v?k}=V (B_i a block in R_j), i.e. R is a resolvable block design RB(v, k, λ). In this paper it will be shown that a sufficient condition for the existence of an RB(v, 8, 7) is that v≡0 (mod 8) and v is nondivisible by 3, 5, 7.     

Further properties of generalized residual designs

The concept of ‘residuation’ is extended so that all ‘generalized residual designs’ (in the sense of Shrinkhande and Singhi) are in fact ‘residual’ with respect to the extended type of residuation. A measure of departure from the usual type of residuation is given in general, and stronger estimates of this measure are given for affine designs.