Partitionable sets, almost partitionable sets and their applications

TL;DR

This paper introduces almost partitionable sets to unify and extend the concept of partitionable sets, enabling new constructions in combinatorial designs and optical orthogonal codes with optimal properties.

Contribution

It defines almost partitionable sets, investigates their existence, and applies them to construct optimal optical orthogonal codes and combinatorial designs.

Findings

Existence results for partitionable and almost partitionable sets

Construction of optical orthogonal codes reaching Johnson bounds

Unified framework for combinatorial design applications

Abstract

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 optical orthogonal codes achieving the Johnson bound or the Johnson bound minus one are constructed.