Related papers: The existence of subspace designs
A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the…
The dimension of a block design is the maximum positive integer $d$ such that any $d$ of its points are contained in a proper subdesign. Pairwise balanced designs PBD$(v,K)$ have dimension at least two as long as not all points are on the…
We discuss a sufficient condition for a space to be filled with an arbitrary finite number of self-similar spaces using a topological concept.
For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior…
In this paper, we prove the existence of a spherical $t$-design formed by adding extra points to an arbitrarily given point set on the sphere and, subsequently, deduce the existence of nested spherical designs. Estimates on the number of…
We prove a metric statement about approximation of a $n$-dimensional linear subspace $A$ in $\mathbb{R}^d$ by $n$-dimensional rational subspaces. We consider the problem of finding a rational subspace $B$ of bounded height $H=H(B)$ for…
We prove several results of the following type: given finite dimensional normed space V there exists another space X with log (dim X) = O(log (dim V)) and such that every subspace (or quotient) of X, whose dimension is not "too small,"…
Guruswami and Xing introduced subspace designs in 2013 to give the first construction of positive rate rank metric codes list-decodable beyond half the distance. In this paper we provide bounds involving the parameters of a subspace design,…
Consider a generic $r$-dimensional subspace of $\mathbb{R}^d$, $r<d$, and suppose that we are only given projections of this subspace onto small subsets of the canonical coordinates. The paper establishes necessary and sufficient…
We give a sufficient condition in order that $n$ closed connected subsets in the $n$-dimensional real projective space admit a common multitangent hyperplane.
We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) log (dim X) = O(log (dim V)) and (2) every…
This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred to as space-filling designs.
We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields…
In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of…
We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying…
Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for…
We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body symmetric about the origin. We also find the order of magnitude of the number of (n-1)-dimensional…
The $q$-analogs of basic designs are discussed. It is proved that the existence of any unknown Steiner structures, the $q$-analogs of Steiner systems, implies the existence of unknown Steiner systems. Optimal $q$-analogs covering designs…
In this survey we give an overview of recent developments on the Quantitative Subspace Theorem. In particular, we discuss a new upper bound for the number of subspaces containing the "large" solutions, obtained jointly with Roberto…
In these notes we show that any projective subspace arrangement can be realized as a generalized star configuration variety. This type of interpolation result may be useful in designing linear codes with prescribed codewords of minimum…