Classification of large partial plane spreads in $PG(6,2)$ and related combinatorial objects

TL;DR

This paper classifies large partial plane spreads in PG(6,2) and uses these results to classify related combinatorial objects such as vector space partitions, MRD codes, and subspace codes, advancing understanding in finite geometry and coding theory.

Contribution

It provides the first complete classification of maximum and near-maximum partial plane spreads in PG(6,2) and derives classifications of related combinatorial structures.

Findings

Classified partial plane spreads of sizes 16 and 17 in PG(6,2)

Classified vector space partitions of type (3^{16} 4^1)

Classified binary MRD codes and subspace codes with specified parameters

Abstract

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6)_2$ and $(7,34,5)_2$.