Partitionable sets, almost partitionable sets and their applications

This paper introduces almost partitionable sets to generalize the known concept of partitionable sets. These notions provide a unified frame to construct $\mathbb{Z}$-cyclic patterned starter whist tournaments and cyclic balanced sampling plans excluding contiguous units. The existences of partitionable sets and almost partitionable sets are investigated. As an application, a large number of maximum or maximal optical orthogonal codes are constructed. These maximal optical orthogonal codes fail to be maximum for just one codeword