On the optimality of block designs

Summary A family of highly efficient designs in the sence of theE-criterion is herein described. These designs have strictly betterE-performance than regular graph designs, yet the off-diagonal entries of theirC-matrix differ by as much as two. Some counterexamples to conjectures in experimental design are then supplied. Asymptotic behavior and equivalence of theA- andD-criteria under a certain condition of uniqueness are analyzed as well.     

On extensions of some block designs

There are some results concerning t-designs in which the number of points in the intersection of two blocks takes less than t values. For example, if t = 2, then the design is symmetric (in such a design, v = b or, equivalently, k = r). In 1974, B. Gross described t-(v, k, l) designs that, for some integer s, 0 < s < t, do not contain two blocks intersecting at exactly s points. Below, it is proved that potentially infinite series of designs from the claim of Gross’ theorem are finite. Gross’ theorem is substantially sharpened.     

On construction of balancedn-ary block designs

Summary A method of construction of balanced ternary designs using affine α-resolvable balanced incomplete block designs is presented.     

On the connectedness of partially balanced block designs

The necessary and sufficient conditions for m-associate partially balanced block (PBB) designs to be connected are given. This generalizes the criterion for m-associate partially balanced incomplete block (PBIB) designs, which has originally been established by Ogawa, Ikeda and Kageyama (1984, Proceedings of the Seminar on Combinatorics and Applications, 248–255, Statistical Publishing Society, Calcutta).This work was partially supported by the Polish Academy of Sciences Grant No. MR I.1-2/2.     

On directed designs with block size five

In at-(v, k,) directed design the blocks are orderedk-tuples and every orderedt-tuple of distinct points occurs in exactly blocks (as a subsequence). We studyt–(v, 5, 1) directed designs witht=3 andt=4. In particular, we construct the first known examples, and an infinite class, of 3-(v, 5, 1) directed designs, and the first known infinite class of 4-(v, 5, 1) directed designs.     

On the existence of super-simple designs with block size 4

Summary We prove that forv = 1 and for allv 1 (mod 3),v 10, there is a (v, 4, 4) design with the property that no triple appears in more than one block. The proof of this result is made more difficult by the non-existence of a GDD (4, 4, 3; 15) with no triple appearing in more than one block. We also show that forv = 1 and for allv 1, 4 (mod 12),v 13, there is a (v, 4, 2) design with this property, and with the additional property that the design is the union of two (v, 4, 1) designs.     

On the incomplete block designs derivable from the association schemes

Blackwelder (1969) has given two methods of constructing balanced incomplete block (BIB) designs from the association matrices of association schemes with two and three associate classes. In this note these two methods are incorporated in a general method, and the existence of a series of BIB designs is shown by the generalized method. In addition, a remark about partially balanced incomplete block (PBIB) designs with respect to the method is made.     

Mixed block designs

Let V_i (i = 1, 2) be a set of size v_i. Let D be a collection of ordered pairs (b₁, b₂) where b_i is a k_i-element subset of V_i. We say that D is a mixed t-design if there exist constants λ _(j,j2), (0 ≤ j_i ≤ k_i, j₁ + j₂ ≤ t) such that, for every choice of a j₁-element subset S₁ of V₁ and every choice of a j₂-element subset S₂ of V₂, there exist exactly λ_(j1,j2) ordered pairs (b₁, b₂) in D satisfying S₁ ⊆ b₁ and S₂ ⊆ b₂. In W. J. Martin [Designs in product association schemes, submitted for publication], Delsarte's theory of designs in association schemes is extended to products of Q-polynomial association schemes. Mixed t-designs arise as a particularly interesting case. These include symmetric designs with a distinguished block and α-resolvable balanced incomplete block designs as examples. The theory in the above-mentioned paper yields results on mixed t-designs analogous to those known for ordinary t-designs, such as the Ray-Chaudhuri/Wilson bound. For example, the analogue of Fisher's inequality gives |D| ≥ v₁ + v₂ − 1 for mixed 2-designs with Bose's condition on resolvable designs as a special case. Partial results are obtained toward a classification of those mixed 2-designs D with |D| = v₁ + v₂ − 1. The central result of this article is Theorem 3.1, an analogue of the Assmus–Mattson theorem which allows us to construct mixed (t + 1 − s)-designs from any t-design with s distinct block intersection numbers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:151–163, 1998     

On representing block designs in terms of their automorphisms

Block designs are analyzed in terms of the structure imposed upon them by their automorphisms. An extension of the notion of a difference set is used to describe necessary and sufficient conditions for the existence of a given automorphism acting on the design. In addition, it is shown that the possible point and block orbit configurations relative to an automorphism acting on a design are rather limited. The development is carried out with a view toward finding unknown designs and studying the automorphism groups of known designs.     

On arcs in path designs of block size 3

For each admissiblev we exhibit a path designP(v, 3, 1) with a spanning set of minimum cardinality and aP(v, 3, 1) with a scattering set of maximum cardinality. Moreover, we study maximal independent sets (or complete arcs in the geometric terminology) having the minimum number of secants, i.e. sets which are both spanning and scattering, and complete arcs with the maximum number of secants.The research for this paper was supported by MURST and GNSAGA-CNR.     

Unitary block designs

Given a projective plane $\text{E}$ over the field of q² elements and a unitary polarity π of $\text{E}$ it is possible to construct the well-known unitary design $\text{U}$ whose points are the absolute points of π and whose blocks are the non-absolute lines of π. A relation of perpendicularity is defined between blocks and it is shown that this relation can be described in terms of the incidence structure of $\text{U}$. The projective plane $\text{E}$ together with the polarity π can then be reconstructed from the design $\text{U}$ in such a way that any automorphism of $\text{U}$ extends to a collineation of $\text{U}$ which commutes with π.     

On covering designs with block size 5 and index 5

LetV be a finite set of order . A (, , ) covering design of index and block size is a collection of -element subsets, called blocks, such that every 2-subset ofV occurs in at least blocks. The covering problem is to determine the minimum number of blocks, (, , ), in a covering design. It is well known that , where [x] is the smallest integer satisfyingx[X]. It is shown here that (, 5, 5)=(, 5, 5) for all positive integers 5 with the possible exception of =24, 28, 56, 104, 124, 144, 164, 184.     

On small packing and covering designs with block size 4

A packing (respectively covering) design of order v, block size k, and index λ is a collection of k-element subsets, called blocks, of a v-set, V, such that every 2-subset of V occurs in at most (at least) λ blocks. The packing (covering) problem is to determine the maximum (minimum) number of blocks in a packing (covering) design. Motivated by the recent work of Assaf [1] [2], we solve the outstanding packing and covering problems with block size 4.