Partial spreads and vector space partitions

TL;DR

This paper reviews the history of partial spreads in finite geometry, introduces vector space partitions as a new framework, and discusses recent improvements in bounds for constant-dimension codes.

Contribution

It provides a modern interpretation of classical results using vector space partitions and summarizes recent advances in bounding the size of partial spreads.

Findings

Vector space partitions improve bounds on partial spreads.

Historical analysis of classical results in finite geometry.

Tutorial introduction to methods from finite geometry and vector space partitions.

Abstract

Constant-dimension codes with the maximum possible minimum distance have been studied under the name of partial spreads in Finite Geometry for several decades. Not surprisingly, for this subclass typically the sharpest bounds on the maximal code size are known. The seminal works of Beutelspacher and Drake \& Freeman on partial spreads date back to 1975, and 1979, respectively. From then until recently, there was almost no progress besides some computer-based constructions and classifications. It turns out that vector space partitions provide the appropriate theoretical framework and can be used to improve the long-standing bounds in quite a few cases. Here, we provide a historic account on partial spreads and an interpretation of the classical results from a modern perspective. To this end, we introduce all required methods from the theory of vector space partitions and Finite Geometry in a tutorial style. We guide the reader to the current frontiers of research in that field, including a detailed description of the recent improvements.