Weighted Subspace Designs from $q$-Polymatroids

TL;DR

This paper extends the Assmus-Mattson theorem to $q$-polymatroids, establishing a new framework for identifying weighted subspace designs from vector space structures.

Contribution

It introduces the characteristic polynomial of $q$-polymatroids and generalizes design theory from matroids to polymatroids and from sets to vector spaces.

Findings

Defined the characteristic polynomial of $q$-polymatroids

Extended design identification methods to vector spaces

Provided properties of the $q$-polymatroid characteristic polynomial

Abstract

The Assmus-Mattson theorem gives a way to identify block designs arising from codes. This result was broadened to matroids and weighted designs. In this work we present a further two-fold generalisation: first from matroids to polymatroids and also from sets to vector spaces. To achieve this, we introduce the characteristic polynomial of a $q$-polymatroid and outline several of its properties.