Sunanda Bagchi

Designs from pairs of finite fields. I. A cyclic unital U (6) and other regular Steiner 2-designs

Abstract A general construction for Steiner 2-designs with prime power block size (and with a point-regular automorphism group) is presented. Its success depends on number-theoretic restrictions on the parameters—these are completely analysed in case of block sizes k ⩽ 11. The new designs constructed include infinitely many cyclic Steiner 2-designs with block size 7. Among them is a cyclic unital U(6), that is, an S(2, 6 + 1, 63 + 1). It is the first example of a unital with non-prime power parameter and the second example of a cyclic unital.

On theE-optimality of certain asymmetrical designs under mixed effects model

AbstractTheE-optimality of the following designs within the class of all proper and connected designs with givenb, k andv under mixed effects model are established.i)A group divisible design with λ2 = λ1 + 1.ii)A group divisible design with λ1 = λ2 + 1 and group size 2.iii)A linked block design.iv)The dual of design (i)v)The dual of design (ii). All these designs are known to satisfy the same optimality property under fixed effects model whenk<v, while the design (i) is known to beE-optimal even whenk>v. From the results proved here, theE-optimality of designs (ii, (iii), (iv) and (v) under fixed effects model in the situation whenk >v also follows.

Optimality and construction of some rectangular designs

We obtain a sufficient condition forE-optimality of equireplicate designs. As an application, we proveE-optimality of certain types of three-class PBIBDs based on rectangular association scheme — in short — rectangular designs. These designs turn out to be highly efficient with respect to theA-criterion as well. We also observe that these designs, though themeselves not regular graph designs (RGDs) are yet strictlyE-better than every competing RGD, wheneverv≥26 andv=2 (mod 4). This provides an infinite series of counter examples to the conjecture of John and Mitchell (1977).We also present two methods of construction of the rectangular designs. Apart from providing infinitely many examples of the designs provedE-optimal in this paper and in Cheng and Constantine (1986), this construction also provides — as a special case — the first known infinite series of most balanced group divisible designs, which were proved optimal with respect to all type 1 criteria by Cheng (1978).

On the optimality of a new class of adjusted orthogonal designs

We define a new class of adjusted orthogonal row–column designs, termed lattice-LBD. These are shown to be E-optimal in the class of all connected row-column designs. Several series of such designs are presented.

A class of non binary unequally replicatedE-optimal designs

In a situation where the given set of parameters (b, k andv) precludes the existence of any known optimal block designs, but an optimal block design is known to exist with parametersb, k andv*>v, a new design is shown to be useful. This (b, k, v) design is obtained from the (b, k, v*) optimal design by collapsing the classes of a suitable paritition of the treatment set (of the latter design) to treatments (of the former). We call the new design a quotient of the original design. Although the quotient is non binary and unequally replicated, it turns out to beE-optimal within the class of all proper and connected designs withb, k andv, provided the replication number of the optimal design we start with is not too large.

On Two-way Designs

Abstract. We discuss two-way designs with special emphasis on the notion of adjusted orthogonality. A restriction on the parameters of adjusted orthogonal two-way designs is obtained. We also exploit the cyclotomy of squares and fourth powers in finite fields to construct an infinite series of adjusted orthogonal two-way designs both of whose marginals are duals of 2-designs.

Construction of group divisible designs and rectangular designs from resolvable and almost resolvable balanced incomplete block designs

Abstract In this paper, we present a general construction of group divisible designs and rectangular designs by utilising resolvable and “almost resolvable” balanced incomplete block designs. As special cases, we obtain the following E-optimal designs: (a) Group divisible (GD) designs with λ2=λ1+1 and (b) Rectangular designs with 2 rows and having λ3=λ2−1=λ1+1. Many of the GD designs are optimal among binary designs with regard to all type 1 criteria.

Two infinite series of E-optimal nested row-column designs

Abstract We construct two infinite series of nested row-column designs having 2 × 4 arrays as blocks and with treatments satisfying a rectangular association scheme with two rows and an odd number (n) of columns. The designs have the following parameters (v = number of treatments, b = number of blocks). Series 1: n ≡ 1 (mod 4), n ⩾ 5, v = 2n, b = n(n − 1) 4 . Series 2: n ≡ 3 (mod 4), n ⩾ 3, v = 2n, b = n(n + 1) 4 . These designs are shown to be E-optimal and highly efficient with respect to A-optimality criterion. Equally important, these optimal designs are very economical in as much as they require as few observations as possible in the set-up.