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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.
This paper is devoted to the study of independent spaces of q-polymatroids. With the aid of an auxiliary q-matroid it is shown that the collection of independent spaces satisfies the same properties as for q-matroids. However, in contrast to q-matroi
A generalization of forming derived and residual designs from $t$-designs to subspace designs is proposed. A $q$-analog of a theorem by Van Trung, van Leijenhorst and Driessen is proven, stating that if for some (not necessarily realizable) parameter
In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Gra{ss}mannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace d
Rudolph (1967) introduced one-step majority logic decoding for linear codes derived from combinatorial designs. The decoder is easily realizable in hardware and requires that the dual code has to contain the blocks of so called geometric designs as c
Let $q$ be a prime power and $Vcong{mathbb F}_q^n$. A $t$-$(n,k,lambda)_q$ design, or simply a subspace design, is a pair ${mathcal D}=(V,{mathcal B})$, where ${mathcal B}$ is a subset of the set of all $k$-dimensional subspaces of $V$, with the prop