Dinesh G. Sarvate

Group divisible designs with three groups and block size four

We present new constructions and results on GDDs with three groups and block size four and also obtain new GDDs with two groups of size nine. We say a GDD with three groups is even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups is always even, always odd, or always mixed. We give new necessary conditions for these families of GDDs and prove that they are sufficient for these three types and for all group sizes except for the minimal case of mixed designs for group size 5t(t>1). In particular, we prove that mixed GDDs allow a maximum difference between indices. We apply the constructions given to show that the necessary conditions are sufficient for all GDDs with three groups and group sizes two, three, and five, and also for group size four with two possible exceptions, a GDD(4,3,4;5,9) and a GDD(4,3,4;7,12).

On c-Bhaskar Rao designs

Abstract We use the theory of balanced incomplete block designs to construct c-Bhaskar Rao designs, also known as generalized balanced matrices, whose rows have constant inner product c. We construct families of c-BRDs with k=3 and c⩾−1 and establish several necessary conditions for the existence of c-BRDs. We also prove that necessary conditions are sufficient for the existence of 1-BRD (v,3,λ), (−1) -BRD(v,3,3), and 2t-BRD(v,3,6t+12s).

Constructions of balanced ternary designs based on generalized Bhaskar Rao designs

Abstract New series of balanced ternary designs and partially balanced ternary designs are obtained. Some of the designs in the series are non-isomorphic solutions for design parameters which were previously known or whose solution was obtained by trial and error, rather than by a systematic method.

General constructions of c-Bhaskar Rao designs and the (c,λ) spectrum of a c-BRD(v,k,λ)

New constructions of Bhaskar Rao designs (BRDs) and new families of c-BRDs are given using the relationship of BRDs to other combinatorial structures. For example, we use different families and their properties in various ways to obtain natural signings of BIBDs in order to construct c-BRDs. We investigate the extreme possibilities for c relative to @l, and k relative to v, and prove the non-existence of an infinite series of designs. We also give a symmetricized version of Ehlichs Hadamard construction.

Coloured designs, new group divisible designs and pairwise balanced designs

Abstract Many new families of group divisible designs, balanced incomplete block designs and pairwise balanced designs can be obtained by using constructions based on coloured designs (CD). This paper gives one such construction in each case together with an existence theorem for coloured designs.

An application of partition of indices to enclosings of triple systems

An enclosing of BIBD(v,3,λ) into BIBD(v+s,3,λ+1) is point-wise minimal if s is the smallest positive integer for which an enclosing is possible. We show that the necessary conditions are sufficient for a minimal point-enclosing of any BIBD(v,3,λ) into BIBD(v + s,3,λ+1) for 1 ≤ λ ≤ 6. Some other general results on enclosings are presented.

Group Divisible Designs with Block Size Five from Clatworthy's Table

ABSTRACT Clatworthys table (Clatworthy, 1973) lists 37 designs with block size 5 where the number of groups is at the most equal to the block size. Mwesigwa, Sarvate, and Zhang have generalized one of these designs in a recent paper (Mwesigwa et al., 2016). In this note we generalize all but one such design listed in the table. As an aside, we prove that group divisible design with intersection pattern (1, 4) does not exist for any n except for n = 4.

GDD ( n , 2 , 4 ; λ 1 , λ 2 ) with equal number of even and odd blocks

We prove that the necessary condition, n ? 0 (mod 3), is sufficient for the existence of GDD( n , 2 , 4 ; 3 , 4 ) except possibly for n = 18 . We prove that necessary conditions for the existence of group divisible designs GDD( n , 2 , 4 ; λ 1 , λ 2 ) with equal number of even and odd blocks are sufficient for GDD ( n , 2 , 4 ; 5 n , 7 ( n - 1 ) ) for all n ? 2 , GDD ( 7 s , 2 , 4 ; 5 s , 7 s - 1 ) for all s , GDD ( 5 t + 1 , 2 , 4 ; 5 t + 1 , 7 t ) for t ? 0(mod 2) and GDD ( 5 t + 1 , 2 , 4 ; 2 ( 5 t + 1 ) , 14 t ) for all t . To complete the existence of such GDDs, one needs to construct two more families: GDD ( 5 t + 1 , 2 , 4 ; 5 t + 1 , 7 t ) for all odd t , and GDD ( 35 s + 21 , 2 , 4 ; 5 s + 3 , 7 s + 4 ) for all positive integers s .

On c-Bhaskar Rao Designs and tight embeddings for path designs

Under the right conditions it is possible for the ordered blocks of a path design PATH(v,k,@m) to be considered as unordered blocks and thereby create a BIBD(v,k,@l). We call this a tight embedding. We show here that, for any triple system TS(v,3), there is always such an embedding and that the problem is equivalent to the existence of a (-1)-BRD(v,3,3), i.e., a c-Bhaskar Rao Design. That is, we also prove the incidence matrix of any triple system TS(v,3) can always be signed to create a (-1)-BRD(v,3,3) and, moreover, the signing determines a natural partition of the blocks of the triple system making it a nested design.

Generalizations of results of Stanton on BIBDs

Abstract We generalize some results of R.G. Stanton concerning intersection numbers and embeddings for balanced incomplete block designs.