Weighted Subspace Designs from q-Polymatroids

Abstract

The Assmus-Mattson Theorem gives a way to identify block designs arising from codes. This result was broadened to matroids and weighted designs by Britz et al. in 2009. 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 study the characteristic polynomial of a q-polymatroid and outline several of its properties. We also derive a MacWilliams duality result and apply this to establish criteria on the weight enumerator of a q-polymatroid for which dependent spaces of the q-polymatroid form the blocks of a weighted subspace design.