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Register a new userWeighted Subspace Designs from $q$-Polymatroids
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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.
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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-matroids, the rank value of an independent space does not agree with its dimension. Nonetheless, the rank values of the independent spaces fully determine the q-polymatroid, and this fact can be exploited to derive a cryptomorphism of q-polymatroids. Finally, the notions of minimal spanning spaces, maximally strongly independent spaces, and bases will be elaborated on.
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 set the derived and residual parameter set are realizable, the same is true for the reduced parameter set. As a result, we get the existence of several previously unknown subspace designs. Some consequences are derived for the existence of large sets of subspace designs. Furthermore, it is shown that there is no $q$-analog of the large Witt design.
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 designs. We construct a $2$-$(6,3,78)_5$ design by computer, which corresponds to a halving $operatorname{LS}_5[2](2,3,6)$. The application of the new recursion method to this halving and an already known $operatorname{LS}_3[2](2,3,6)$ yields two infinite two-parameter series of halvings $operatorname{LS}_3[2](2,k,v)$ and $operatorname{LS}_5[2](2,k,v)$ with integers $vgeq 6$, $vequiv 2mod 4$ and $3leq kleq v-3$, $kequiv 3mod 4$. Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$.
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 codewords. Peterson and Weldon (1972) extended Rudolphs algorithm to a two-step majority logic decoder correcting the same number of errors than Reeds celebrated multi-step majority logic decoder. Here, we study the codes from subspace designs. It turns out that these codes have the same majority logic decoding capability as the codes from geometric designs, but their majority logic decoding complexity is sometimes drastically improved.
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 property that each $t$-dimensional subspace of $V$ is contained in precisely $lambda$ elements of ${mathcal B}$. Subspace designs are the $q$-analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group ${rm Aut}({mathcal D})$ acts transitively on ${mathcal B}$. It is shown here that if $tgeq 2$ and ${mathcal D}=(V,{mathcal B})$ is a block-transitive $t$-$(n,k,lambda)_q$ design then ${mathcal D}$ is trivial, that is, ${mathcal B}$ is the set of all $k$-dimensional subspaces of $V$.
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