The existence of subspace designs

TL;DR

This paper proves the existence of subspace designs with arbitrary parameters in large enough spaces, resolving a long-standing open problem from the 1970s and providing an approximate count formula.

Contribution

It establishes the existence of subspace designs for any parameters under certain conditions and offers an approximate enumeration formula.

Findings

Existence of subspace designs for large enough spaces

Resolution of a 1970s open problem

Approximate formula for counting such designs

Abstract

We prove the existence of subspace designs with any given parameters, provided that the dimension of the underlying space is sufficiently large in terms of the other parameters of the design and satisfies the obvious necessary divisibility conditions. This settles an open problem from the 1970s. Moreover, we also obtain an approximate formula for the number of such designs.