Donald L. Kreher

Covering Arrays of Strength Three

A covering array of size N, degree k, order v and strength t is a k × N array with entries from a set of v symbols such that in any t × N subarray every t × 1 column occurs at least once. Covering arrays have been studied for their applications to drug screening and software testing. We present explicit constructions and give constructive upper bounds for the size of a covering array of strength three.

The existence of simple 6-(14,7,4) designs

Abstract A cyclic 5-(13, 6, 4) design is constructed and is extended to a simple 6-(14, 7, 4) design via a theorem of Alltop. This design is the smallest possible nontrivial simple 6-design that can exist. Both have full automorphism group cyclic of order 13.

Super-simple (υ, 5, 2)-designs

In this paper we study the spectrum of super-simple (ν, 5, 2)-designs. We show that a super-simple (ν, 5, 2)-design exists if and only if ν≡1 or 5 (mod 10), ν ≠ 5, 15, except possibly when ν∈ {75, 95, 115, 135, 195, 215, 231, 285, 365, 385, 515}.

Small group-divisible designs with block size four

In this paper we study the group-divisible designs with block size four on at most 30 points. For all but three of the possible group types, we determine the existence or non-existence of the design.

6-regular Cayley graphs on abelian groups of odd order are hamiltonian decomposable

Alspach conjectured that any 2k-regular connected Cayley graph on a finite abelian group A has a hamiltonian decomposition. In this paper, the conjecture is shown true if k=3, and the order of A is odd.

Concerning Difference Matrices

Several new constructions for difference matrices are given. One classof constructions uses pairwise balanced designs to develop newdifference matrices over the additive group of GF (q). A second class of constructions gives difference matrices overgroups whose orders are not (necessarily) prime powers.

On the covering of t -sets with ( t +1)-sets: C(9,5,4) and C(10,6,5)

Abstract A (υ, k , t ) covering system is a pair ( X , B ) where X is a υ-set of points and B is a family of k -subsets, called blocks, of X such that every t -subset of X is contained in at least one block. The minimum possible number of blocks in a (υ, k , t ) covering system is denoted by C (υ, k , t ). It is proven that there are exactly three non-isomorphic systems giving C (9, 5, 4) = 30, and a unique system giving C (10, 6, 5) = 50.

A note on {4}-GDDs of type 2 10

Motivated by a problem on resolvable designs we compute all non-isomorphic group divisible designs of type 210 with block-size 4 and show that none has a parallel class.

Partial Steiner triple systems with equal-sized holes

Abstract The existence of group divisible designs of type u r 1 t with block size three is completely settled for all values of u , r , and t .

Pooling, lattice square, and union jack designs

Simplified pooling designs employ rows, columns, and principal diagonals from square and rectangular plates. The requirement that every two samples be tested together in exactly one pool leads to a novel combinatorial configuration: The union jack design. Existence of union jack designs is settled affirmatively whenever the ordern is a prime andn≡3 (mod 4).