Group divisible designs with block size four and two groups

We give some constructions of new infinite families of group divisible designs, GDD(n,2,4;λ₁,λ₂), including one which uses the existence of Bhaskar Rao designs. We show the necessary conditions are sufficient for 3?n?8. For n=10 there is one missing critical design. If λ₁>λ₂, then the necessary conditions are sufficient for . For each of n=10,15,16,17,18,19, and 20 we indicate a small minimal set of critical designs which, if they exist, would allow construction of all possible designs for that n. The indices of each of these designs are also among those critical indices for every n in the same congruence class mod 12.     

Weak amicable T‐matrices and Plotkin arrays

We introduce the class of weak amicable T‐matrices and use it to construct a class of orthogonal designs, for p = 1 and for p a prime power ≡ 3 (mod 4), and all odd q, q ≤ 21. This class includes new Plotkin arrays of order 24, 40, 56 and for the first time, of orders 8q, q ∈ {9,11,13,15,17,19,21}. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 44–52, 2008     

A Note on the Construction of Orthogonal Latin Hypercube Designs

Latin hypercube designs have been found very useful for designing computer experiments. In recent years, several methods of constructing orthogonal Latin hypercube designs have been proposed in the literature. In this article, we report some more results on the construction of orthogonal Latin hypercubes which result in several new designs.     

Partial Geometric Difference Families

We introduce the notion of a partial geometric difference family as a variation on the classical difference family and a generalization of partial geometric difference sets. We study the relationship between partial geometric difference families and both partial geometric designs and difference families, and show that partial geometric difference families give rise to partial geometric designs. We construct several infinite families of partial geometric difference families using Galois rings and the cyclotomy of Galois fields. From these partial geometric difference families, we generate a list of infinite families of partial geometric designs and directed strongly regular graphs.     

幻方的几个充要条件

建立等重方阵的概念,并运用它阐述幻方、2次幻方、3次幻方及中心对称互补幻方的充要条件,根据这些结论构作出一些8阶、9阶2次中心对称互补幻方.     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

Completing Latin squares: Critical sets II

It is shown that each critical set in a Latin square of order n > 6 has to have at least empty cells. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 77–83, 2007     

关于五点图的多部图设计

讨论了完全多部图的G设计的存在性,其中G是五点四边图和五点五边图。并给出其存在谱。     

Arrays for orthogonal designs

A set of square real matrices is said to be amicable if for some permutation σ of the set . An infinite number of arrays which are suitable for any amicable set of eight circulant matrices are introduced. Applications include new classes of orthogonal designs. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 166–173, 2000     

Coloring BIBDs with block size 4

It is shown that for each integer m ≥ 1 there exists a lower bound, v_m, with the property that for all v ≥ v_m with v ≡ 1, 4 (mod 12) there exists an m-chromatic S(2, 4, v) design. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 403–409, 1998