Generalized vector space partitions

TL;DR

This paper generalizes the concept of vector space partitions to include t-dimensional subspaces, exploring their properties and connections to subspace codes, with some problems remaining open even in the simplified case.

Contribution

It introduces the notion of vector space t-partitions, extending classical partitions and analyzing their properties and relationships to subspace coding theory.

Findings

Established properties of vector space t-partitions

Connected t-partitions to subspace codes

Identified open problems for q=1 case

Abstract

A vector space partition $\mathcal{P}$ in $\mathbb{F}_q^v$ is a set of subspaces such that every $1$-dimensional subspace of $\mathbb{F}_q^v$ is contained in exactly one element of $\mathcal{P}$. Replacing "every point" by "every $t$-dimensional subspace", we generalize this notion to vector space $t$-partitions and study their properties. There is a close connection to subspace codes and some problems are even interesting and unsolved for the set case $q=1$.