On GMW Designs and Cyclic Hadamard Designs

We generalise results of Jackson concerning cyclic Hadamard designs admitting SL(2,2ⁿ) as a point transitive automorphism group. The generalisation concerns the designs of Gordon, Mills and Welch and we characterise these as designs admitting GM(m,qⁿ) acting in a certain way. We also generalise a construction given by Maschietti, using hyperovals, of cyclic Hadamard designs, and characterise these amongst the designs of Gordon, Mills and Welch.     

Cube Designs

A cube design of order v, or a CUBE(v), is a decomposition of all cyclicly oriented quadruples of a v‐set into oriented cubes. A CUBE(v) design is unoriented if its cubes can be paired so that the cubes in each pair are related by reflection through the center. A cube design is degenerate if it has repeated points on one of its cubes, otherwise it is nondegenerate. We show that a nondegenerate CUBE(v) design exists for all integers , and that an unoriented nondegenerate CUBE(v) design exists if and only if and or . A degenerate example of a CUBE(v) design is also given for each integer .     

Perfect Cayley Designs as Generalizations of Perfect Mendelsohn Designs

We introduce the concept of a Perfect Cayley Design(PCD) that generalizes that of a Perfect Mendelsohn Design (PMD) as follows. Given anadditive group H, a (v, H, 1)-PCDis a pair whereX is a v-set and isa set of injective maps fromH toX with the property that for any pair (x,y)of distinct elements of X and any h H - {0} there is exactly one B such that B(h')=x, B(h')=yandh'-h'=h for suitable h',h' H.It is clear that a (v,Z_k,1)-PCD simply is a(v, k, 1)-PMD.This generalization has concretemotivations in at least one case. In fact we observe thattriplewhist tournaments may be viewed as resolved(v,Z ₂ ² ,1)-PCD's but not, in general, as resolved(v, 4, 1)-PMD's.We give four composition constructionsfor regular and 1-rotational resolved PCD's. Two of them make use of differencematrices and contain, asspecial cases, previous constructions for PMD's by Kageyama andMiao [15] and for Z-cyclic whist tournaments by Anderson,Finizio and Leonard [5]. The other two constructions succeed wheresometimes difference matrices fail and their applications allow us to get new PMD's, new Z-cyclic directed whist tournaments and newZ-cyclic triplewhist tournaments.The whist tournaments obtainable with the last twoconstructions are decomposable into smaller whist tournaments.We show this kind of tournaments useful in practice whenever, at theend of a tournament, some confrontations between ex-aequo players areneeded.     

Symmetric Graph Designs

We introduce the notion of a symmetric graph design which is a common generalization of symmetric BIBDs and of orthogonal double covers. Revised: January 21, 1999     

λKv的最大K2,3填充设计和最小K2,3覆盖设计

对于一个有限简单图G,λK_v的G-设计(G-填充,G-覆盖),记为(v,G,λ)-GD((v,G,λ)-PD,(v,G,λ)-CD),是一个(X,B),其中X是K_v的顶点集,B是K_v的子图族,每个子图(称为区组)均同构于G,且K_v中任一边都恰好(最多,至少)出现在B的λ个区组中.一个填充(覆盖)设计称为是最大(最小)的,如果没有其它的这种填充(覆盖)设计具有更多(更少)的区组.本文对于λ>1确定了(v,K_2,3,λ)-GD的存在谱,并对任意λ构造了λK_v的最大K_2,3-填充设计和最小K_2,3-覆盖设计.     

Exponentially Many Hadamard Designs

If there is a Hadamard design of order n, then there are at least 2^{8n−16−9log n} non-isomorphic Hadamard designs of order 2n. Mathematics Subject Classificaion 2000: 05B05     

混料模型的混合最优设计

最优设计是试验设计中必不可少的一种设计方法,混合最优设计是二步最优设计中一种常见的应用。本文以混料模型为基础,提出一种新的寻求混合最优设计的方法,并以此方法解决了混料模型的混合最优设计问题。     

Orthogonally Resolvable Matching Designs

An orthogonally resolvable matching design OMD$(n,k)$ is a partition of the edges of the complete graph $K_n$ into matchings of size $k$, called blocks, such that the blocks can be resolved in two different ways. Such a design can be represented as a square array whose cells are either empty or contain a matching of size $k$, where every vertex appears exactly once in each row and column. In this paper we show that an OMD$(n,k)$ exists if and only if $n\equiv 0(mod2k)$ except when $k=1$ and $n=4$ or $6$.     

Cocyclic Development of Designs

We present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G × G. Negacyclic and -cyclic development are both special cases of cocyclic development.Techniques of design construction using the group ring, arising from difference set methods, also apply to cocyclic designs. Important classes of Hadamard matrices and generalized weighing matrices are cocyclic.We derive a characterization of cocyclic development which allows us to generate all matrices which are cocyclic over G. Any cocyclic matrix is equivalent to one obtained by entrywise action of an asymmetric matrix and a symmetric matrix on a G-developed matrix. The symmetric matrix is a Kronecker product of back -cyclic matrices, and the asymmetric matrix is determined by the second integral homology group of G. We believe this link between combinatorial design theory and low-dimensional group cohomology leads to (i) a new way to generate combinatorial designs; (ii) a better understanding of the structure of some known designs; and (iii) a better understanding of known construction techniques.