Balanced 6 × 6 designs for 4 equally replicated treatments

The parameters 4-(12, 6, 6) and related t-designs

It is shown that a 4-(12,6,6) design, if it exists, must be rigid. The intimate relationship of such a design with 4-(12,5,4) designs and 5-(12,6,3) designs is presented and exploited. In this endeavor we found: (i) 30 nonisomorphic 4-(12,5,4) designs; (ii) all cyclic 3-(11,5,6) designs; (iii) all 5-(12,6,3) designs preserved by an element of order three fixing no points and no blocks; and (iv)...

Control Balanced Designs Involving Sequences of Treatments

In search of 4-(12, 6, 4) designs: Part III

A 4-(12, 6, 4) design that is not also a 5-(12, 6, 1) design must have at least one pair of blocks with five points in common. It is shown that there are just nine non-isomorphic such designs; so, including the 5-(12, 6, 1) design, there are ten 4-(12, 6, 4) designs. These designs are characterised by the orders of their automorphism groups and they all contain a 4-(11, 5, 1) design.

In search of 4-(12, 6, 4) designs: Part I

Martin J. Sharry and Anne Penfold Street, Centre for Combinatorics, Department of Mathematics, The University of Queensland, Brisbane 4072, Australia. As a first step towards finding all 4-(12, 6, 4) designs which are not 5-(12, 6, 1) designs, it is shown that if such a design has a pair of blocks with five points in common, then there is a unique way of assigning the replicas of the seven poin...

In search of 4-(12, 6, 4) designs: Part II

In a 4-(12, 6, 4) design a block is either disjoint from one other block or it has five points in common with one other block. For a 4-(12, 6, 4) design with a pair of blocks of the second type it is shown that· another thirty blocks of the design can be completed in a unique way and these thirty blocks contain a copy of a 3-(10, 4, 1) design.