A common generalization of hypercube partitions and ovoids in polar spaces

The article introduces the concept of generalized ovoids in polar spaces, a natural q-analog of subcube partitions in hypercubes. It provides a detailed study of these structures, including their definitions, constructions, and asymptotic non-existence results. The main goal is to determine the minimum size of tight irreducible subcube partitions and other natural extremal questions. The authors also discuss the connection between these concepts and complexity theory, extending previous work to include hypercubes with larger alphabets and linear subspaces. The results are presented through a series of theorems and constructions, offering new insights into the field of finite geometry and combinatorics.