Regular Group Divisible Designs Research Articles

A Survey on Partially Resolvable Solutions of Regular Group Divisible Designs

A Survey on Partially Resolvable Solutions of Regular Group Divisible Designs

Balanced Arrays Through Symmetrical Regular Group Divisible Designs

An alternative and simple method of construction of balanced array, orthogonal and proportional frequency main effect plans and asymmetrical non-orthogonal main effect plans using incidence matrix of symmetrical regular GD designs has been proposed. The above plans can be constructed by symmetrical regular GD designs only when p is a positive integer.

A new regular group divisible design

A regular group divisible design with parameters: v=b=39, r=k=9, λ1=0, λ2=2, m=13, n=3 is obtained using balanced generalized Weighing matrix over a dihedral group of order 6.

Non-validity of affine [formula omitted]-resolvability in regular group divisible designs

The concept of affine α -resolvability has been discussed for block designs in literature since 1942 for α = 1 and in particular since 1963 for α ⩾ 2 . Among group divisible (GD) designs, affine α -resolvable designs are known for both classes of singular GD and semi-regular GD designs. However, no example has been found for an affine α -resolvable regular GD design in literature. In this paper, the validity of such concept will be disproved for regular GD designs in general. Thus the regular GD design does not possess any property of the affine α -resolvability.

On Non-Existence of Affine α-Resolvable Triangular Designs

The existence on affine αresolvability with some properties has been discussed for block designs in literature since 1942 for a =1 and in particular since 1963 for a ≥ 2. Recently, Kageyama (2007) disproved the validity of such concept for regular group divisible designs. A similar problem for 2-associate triangular designs will be tackled here. No example has been found for an affine αresolvable triangular design for any a in literature. In this paper, when a = 1,2, the non-existence of an affine αresolvable triangular design will be shown completely, whereas, when a ≥ 3, partial non-existence results are provided. AMS Subject Classification: Primary 62K10, Secondary 05B05.

Old and new designs via difference multisets and strong difference families

Let X =((x1,1,x1,2,…,x1,k),(x2,1,x2,2,…,x2,k),…,(xt,1,xt,2,…,xt,k)) be a family of t multisets of size k defined on an additive group G. We say that X is a t-(G,k,μ) strong difference family (SDF) if the list of differences (xh,i-xh,j∣h=1,…,t;i≠ j) covers all of G exactly μ times. If a SDF consists of a single multiset X, we simply say that X is a (G,k,μ) difference multiset. After giving some constructions for SDF's, we show that they allow us to obtain a very useful method for constructing regular group divisible designs and regular (or 1-rotational) balanced incomplete block designs. In particular cases this construction method has been implicitly used by many authors, but strangely, a systematic treatment seems to be lacking. Among the main consequences of our research, we find new series of regular BIBD's and new series of 1-rotational (in many cases resovable) BIBD's.

Old and new designs via difference multisets and strong difference families

Let X =((x1,1,x1,2,…,x1,k),(x2,1,x2,2,…,x2,k),…,(xt,1,xt,2,…,xt,k)) be a family of t multisets of size k defined on an additive group G. We say that X is a t-(G,k,μ) strong difference family (SDF) if the list of differences (xh,i-xh,j∣h=1,…,t;i≠ j) covers all of G exactly μ times. If a SDF consists of a single multiset X, we simply say that X is a (G,k,μ) difference multiset. After giving some constructions for SDF's, we show that they allow us to obtain a very useful method for constructing regular group divisible designs and regular (or 1-rotational) balanced incomplete block designs. In particular cases this construction method has been implicitly used by many authors, but strangely, a systematic treatment seems to be lacking. Among the main consequences of our research, we find new series of regular BIBD's and new series of 1-rotational (in many cases resovable) BIBD's.

Constructions of nested group divisible designs

Several patterned methods of construction of nested group divisible (NGD) designs, starting from self-complementary group divisible (GD) designs and Hadamard matrices or GD designs, are given with some series of NGD designs. As a by-product, a new regular GD design is found.

Spreads and group divisible designs

By removing the components of at-spread\(S\) of a finite projective spacePG(d, q) from each hyperplane ofPG(d, q), the blocks of a regular group divisible design\(\mathcal{G}(S)\) are obtained We characterize geometrict-spreads as thoset-spreads\(S\) which are such that the dual of\(\mathcal{G}(S)\) is also a group divisible design.

Efficiency of connected binary block designs when a single observation is unavailable

In this paper the problem of finding the design efficiency is considered when a single observation is unavailable in a connected binary block design. The explicit expression of efficiency is found for the resulting design when the original design is a balanced incomplete block design or a group divisible, singular or semiregular or regular with λ1>0, design. The efficiency does not depend on the position of the unavailable observation. For a regular group divisible design with λ1>0, the efficiency depends on the position of the unavailable observation. The bounds, both lower and upper, on the efficiency are given in this situation. The efficiencies of designs resulting from a balanced incomplete block design and a group divisible design are in fact high when a single observation is unavailable.

Robustness group divisible designs

Robustness of group divisible (GD) designs is investigated, when one block is lost, in terms of efficiency of the residual design. The exact evaluation of the efficiency can be made for singular GD and semi-regular GD designs as ell as regular GD designs with λ1 = 0. In a regular GD design with λ1 > 0, the efficiency may depend upon the lost block and sharp upper and lower bounds on the efficiency are presented. The investigation shows that GD designs are fairly robust in terms of efficiency. As a special case, we can also show the robustness of balanced incomplete block design when one block is lost.

A Method of Construction of Regular Group Divisible Designs

A Method of Construction of Regular Group Divisible Designs

A method of construction of regular group divisible designs

A method of construction of regular group divisible designs

Symmetry and classification of certain regular group divisible designs

In this paper it is shown that a regular group divisible (GD) design, with parametersv, b, r, k, ? 1, ?2 satisfyingrk ? ?2 v + 1 and? 2 = ?1 + 1, must be symmetric (i.e.,v + b). Furthermore, the parameters of such symmetric regular GD designs can be expressed in terms of only two integral parameters.

A construction of group divisible designs

A method of construction of group divisible (GD) designs from a resolvable balanced incomplete block (BIB) design is given with new families. Within the scope of practical ranges of parameters, new non-isomorphhic solutions for regular GD designs are produced.

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

Abstract 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, λ 1 =0, group size 2 and υ groups.

Balanced arrays and main-effect plans from regular group-divisible designs

Abstract A method of constructing balanced arrays of strength two and corresponding main-effect fractional factorial plans from some regular group-divisible designs is presented. Main-effect fractional factorial plans for three-level factorial experiments are constructed. The plans have reasonably high efficiencies for different single degree of freedom effects.

Some families of group divisible designs

Abstract Generalizing methods of constructions of Hadamard group divisible designs due to Bush (1979), some new families of semi-regular or regular group divisible designs are produced. Furthermore, new nonisomorphic solutions for some known group divisible designs are given, together with useful group divisible designs not listed in Clatworthy (1973).

Construction of Regular Group Divisible Designs

Construction of Regular Group Divisible Designs

Construction of regular group divisible designs

Construction of regular group divisible designs