We show that the necessary conditions λ = 0 (mod IGI), λ(v-l)=0 (mod2), λv(v 1) = [0 (mod 6) for IGI odd, (0 (mod 24) for IGI even, are sufficient for the existence of a generalized Bhaskar Rao design GBRD(v,b,r,3,λ;G) for the elementary abelian group G, of each order IGI. Disciplines Physical Sciences and Mathematics Publication Details Seberry, J, Generalized Bhaskar Rao designs with block si...

A colouring of a 4-cycle system (V,B) is a surjective mapping φ : V → Γ. The elements of Γ are colours and, for each i ∈ Γ, the set Ci = φ−1(i) is a colour class. If |Γ| = m, we have an m-colouring of (V,B). For every B ∈ B, let φ(B) = {φ(x) | x ∈ B}. We say that a block B is equitably coloured if either |φ(B)∩Ci| = 0 or |φ(B)∩Ci| = 2 for every i ∈ Γ. Let F(n) be the set of integersm such that ...

Stufken (1993) first introduced cyclic polygonal designs (CPDs) (in terms of cyclic balanced sampling plans excluding adjacent units (CBSAs)) for joint distance α ≥ 2. Wei (2002) first suggested the use of Langford sequence for the existence and construction of CPDs with block size k = 3 and λ = 1 for arbitrary α . Zhang and Chang (2005b) used Langford and extended Langford sequence, and constr...

Let G be a group with socle a simple group of Lie type defined over the finite field with q elements where q is a power of the prime p. Suppose that G acts transitively upon the lines of a linear space S. We show that if p is significant then G acts flag-transitively on S and all examples are known. MSC(2000): 20B25, 05B05. 1 Background and statement of result A linear space S is an incidence s...

A Kirkman triple system of order v, whose point set is consecutively numbered, is called smooth, if for each parallel class there is a number d, such that each triple of the parallel class forms an arithmetic progression modulo v with common difference d. It will be shown that smooth Kirkman triple systems of order v exist, if and only if v is a power of 3. 1 Smooth Kirkman triple systems The f...

A colouring of a 4-cycle system (V,B) is a surjective mapping φ : V → Γ. The elements of Γ are colours. If |Γ| = m, we have an m-colouring of (V,B). For every B ∈ B, let φ(B) = {φ(x)|x ∈ B}. There are seven distinct colouring patterns in which a 4-cycle can be coloured: type a (××××, monochromatic), type b (×××2, two-coloured of pattern 3 + 1), type c (× × 22, two-coloured of pattern 2 + 2), ty...

For all ‘reasonable’ finite t, k and s we construct a t-(א0, k, 1) design and a group of automorphisms which is transitive on blocks and has s orbits on points. In particular, there is a 2-(א0, 4, 1) design with a block-transitive group of automorphisms having two point orbits. This answers a question of P. J. Cameron and C. E. Praeger. The construction is presented in a purely combinatorial wa...

A Metis design is one for which v = r + k + 1. This paper deals with Metis designs that are quasi-residual. The parameters of such designs and the corresponding symmetric designs can be expressed by Fibonacci numbers. Although the question of existence seems intractable because of the size of the designs, the nonexistence of corresponding di¤erence sets can be dealt with in a substantive way. W...

A triple system of order v ≥ 3 and index λ is faithfully enclosed in a triple system of order w ≥ v and index μ ≥ λ when the triples induced on some v elements of the triple system of order w are precisely those from the triple system of order v. When λ = μ, faithful enclosing is embedding; when λ = 0, faithful enclosing asks for an independent set of size v in a triple system of order w. When ...

For each prime power q = 1 (mod 12), there is a triple system of order q whose automorphism group is transitive on unordered pairs. The object of this paper is to study these systems. This is done by analyzing how pairs of elements are linked. The linkage of a and b consists of a triple (a, b, c) and of some cycles in which adjacent pairs of elements form triples alternately with a and with b. ...