Skolem-type sequences can be useful in constructing various types of designs, for example in constructing Steiner triple systems and cyclic partial triple systems.

A k-extended q-near Skolem sequence of order n is an integer sequence (s₁, ...,s_2n-1) with s_k=0 such that for each j Î {1,2, ..., n} \{q}, there exists a unique i with s_i=s_i+j=j. It is straightforward to show that such a sequence exists only if

(1) n º 0, 1 mod 4 and q and k have the same parity, or
(2) n º 2, 3 mod 4 and q and k have opposite parity.

A complete multipartite graph K(a₁,a₂,...,a_n) has its vertices partitioned into n parts or "partite sets" of size a_i, 1\leqslant i\leqslant n, and any pair of vertices is joined by an edge if and only if the vertices lie in different partite sets.

A k-cycle decomposition of G = K(a₁,a₂,...,a_n) is a partition of all the edges of G into k-cycles. The decomposition is said to be `gregarious' if every possible k-cycle in the decomposition has all its k vertices lying in different partite sets (so necessarily the cycle length k does not exceed the number of parts n).

An update on the present state of play regarding existence of gregarious k-cycle systems will be given.

Joint with Benjamin R. Smith.

A 2-(v,k,l) packing design, (V,B), is a set V (with elements called points) and a collection B of k-subsets of V (called blocks) with the property that every unordered pair of points occurs in at most l blocks. We denote the maximum possible size of B by D_l(v,k,2) and call it the packing number for these parameters. We are interested in finding either the exact value of D_l(v,k,2) or a good lower bound on it.

I will give an update on the exact values of D_l(v,5,2).

I will also talk about some new results on improving the known bounds on the size of constant weight codes (packings with l = 1) by using optimization.

We consider the problem of determining necessary and sufficient conditions for the existence of a decomposition of the complete multigraph lK_m into cycles of length k (also called a (lK_m, C_k)-design or a k-cycle system of lK_m). This problem remains open in general. Notably, necessary and sufficient conditions are known for l = 1 or 2 (Alspach, Gavlas, Sajna, Verrall), but have not been determined for greater values of l. In this talk, we discuss progress towards the case l = 3.

It is shown that if F₁, F₂, ..., F_t are bipartite 2-regular graphs of order n and a₁, a₂, ..., a_t are non-negative integers such that a₁ + a₂ + ¼+ a_t = [(n-2)/2], a₁ ³ 3 is odd, and a_i is even for i = 2, 3, ..., t, then there exists a 2-factorisation of K_n-I in which there are exactly a_i 2-factors isomorphic to F_i for i = 1, 2, ..., t. Taking t=1 this result completes the solution of the Oberwolfach problem for any collection of even sized cycles.

This is joint work with Darryn Bryant, The University of Queensland.

In cryptography (as well as other areas, I'm sure), the effective (ab)use of random bits is of great importance. In this talk, we consider an expansion function f : {0,1}^m ® {0,1}ⁿ (n > m) with the property that, given the uniform distribution U_m on input strings, the projection of the output f(U_m) onto any t coordinates has min-entropy at least l. For example, for l = t, this is just a binary orthogonal array of strength t with n factors and 2^m runs. Our goal is to significantly beat the Rao bound by allowing l to drop below t.

In this talk, which is joint work with Matt Houde of EMC Corporation, I will give some preliminary bounds and constructions. Hopefully, I can motivate some experts to look into this question.

Consider a projective plane over a field F. One interesting question is: given two triangles in perspective from a point V, under what circumstances does V lie on the Desargues axis? We consider Desargues' theorem and some related configurations.

This is joint work with Prof. Aiden Bruen, Department of Electrical and Computer Engineering, the University of Calgary.

A Skolem sequence of order n is a sequence S_n = (s₁,s₂,...,s_2n) of 2n integers containing each of the symbols 1,2,...,n exactly twice, such that two occurrences of the integer j Î {1,2,...,n} are separated by exactly j-1 symbols. We prove, with few possible exceptions, that there exists two Skolem sequences of order n with 0,1,2,...,n-3 or n pairs in common. Using this result, we determine, with few possible exceptions the fine structure of a cyclic three-fold triple systems, for v º 1,7 mod 24. We also determine, with few exceptions, the fine structure of a cyclic four-fold triple systems, for v º 1,7 mod 24. Then, we extend these results to the fine structure of a l-fold directed triple system and a l-fold Mendelsohn triple system. We also determine, the number of possible repeated base blocks in a cyclic two-fold triple system, a directed triple system and a Mendelsohn triple system, for v º 1,3 mod 6.

An octahedral design of order v, or ocv, is a decomposition of all oriented triples on v points into oriented octahedra. Hanani settled the existence of these designs in the unoriented case. We show that an ocv exists if and only if v º 1,2,6 mod 8 (the admissible numbers), and moreover the constructed ocv are indecomposble, i.e., the octahedra cannot be paired into mirror images. We show that an ocv with a subdesign ocu exists if and only if v and u are admissible and v ³ u+4.

This is joint work with Prof. V. Linek.