
Skolemtype sequences can be useful in constructing various types of designs, for example in constructing Steiner triple systems and cyclic partial triple systems.
A kextended qnear Skolem sequence of order n is an integer sequence (s_{1}, ...,s_{2n1}) 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
A complete multipartite graph K(a_{1},a_{2},...,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 kcycle decomposition of G = K(a_{1},a_{2},...,a_{n}) is a partition of all the edges of G into kcycles. The decomposition is said to be `gregarious' if every possible kcycle 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 kcycle 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 ksubsets 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 kcycle 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 F_{1}, F_{2}, ..., F_{t} are bipartite 2regular graphs of order n and a_{1}, a_{2}, ..., a_{t} are nonnegative integers such that a_{1} + a_{2} + ¼+ a_{t} = [(n2)/2], a_{1} ³ 3 is odd, and a_{i} is even for i = 2, 3, ..., t, then there exists a 2factorisation of K_{n}I in which there are exactly a_{i} 2factors 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 minentropy 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} = (s_{1},s_{2},...,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 j1 symbols. We prove, with few possible exceptions, that there exists two Skolem sequences of order n with 0,1,2,...,n3 or n pairs in common. Using this result, we determine, with few possible exceptions the fine structure of a cyclic threefold triple systems, for v º 1,7 mod 24. We also determine, with few exceptions, the fine structure of a cyclic fourfold triple systems, for v º 1,7 mod 24. Then, we extend these results to the fine structure of a lfold directed triple system and a lfold Mendelsohn triple system. We also determine, the number of possible repeated base blocks in a cyclic twofold 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.
Costas arrays arise in sonar and radar applications and they also closely related to a few other combinatorial designs. Originally an m ×m Costas array is defined as an m ×m permutation matrix (that is, a square matrix with precisely one 1 in each row and column and all other entries 0) for which all the vectors joining the pairs of 1's are distinct. It can also be defined in terms of a permutation such that each row in the difference triangle contains distinct entries. In this talk, I will discuss basic constructions of Costas arrays and a weaker notion of Costas arrays: permutations with distinct difference property (DDP permutations). In particular, I will address some issues related to a construction proposed by Batten and Sane in 2003 on DDP permutations.