Cyclic block designs from Skolem-type sequences

Abstract

M. Colbourn and R. Mathon [45] asked: "Can Skolem's partitioning problems be generalized to yield cyclic BIBD(v; 4; 1)?". Rosa [76] asked: \What is the format of Skolem-type sequences that leads to cyclic BIBD(v; k; λ) for k [greater than or equal to] 4?". In this thesis, we will address these two questions. We introduce new Skolem-type sequences and then we use them to construct new cyclic BIBD(v; k; λ) for k [greater than or equal to] 3. Specifically, we use Skolem-type sequences to construct new cyclic BIBD(v; 3; λ) for all admissible orders v and λ. We use Skolem-type sequences to construct new cyclic BIBD(v; k; λ) for k [greater than or equal to] 4 and every v coprime with 6. We provide a complete set of examples of Skolem partitions that induce one cyclic BIBD(v; 4; λ) for every admissible class. We also use some known results and relative difference families to construct new cyclic BIBD(v; 4; λ) for infinite values of v. Moreover, we use Skolem-type sequences to construct cyclic, simple, and indecomposable BIBD(v; 3; 3) for every v with some possible exceptions for v = 9 and v = 24c + 9, c 4. We also construct infinitely many cyclically indecomposable but decomposable BIBD(v; 3; 4) for some orders v. Finally, we have many examples of simple and super-simple cyclic designs coming from Skolem-type sequences that produce optical orthogonal codes.