Related papers: Strong blocking sets and minimal codes from expand…
In this paper we consider how the strong-coupling scale, or perturbative cutoff, in a multi-gravity theory depends upon the presence and structure of interactions between the different fields. This can elegantly be rephrased in terms of the…
This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only…
We construct constant-sized ensembles of linear error-correcting codes over any fixed alphabet that can correct a given fraction of adversarial erasures at rates approaching the Singleton bound arbitrarily closely. We provide several…
Let G be a complete convex geometric graph on 2m vertices, and let F be a family of subgraphs of G. A blocker for F is a set of edges, of smallest possible size, that meets every element of F. In [C. Keller and M. A. Perles, On the smallest…
For any constants $d\ge 1$, $\epsilon >0$, $t>1$, and any $n$-point set $P\subset\mathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$ having $O(n\log^2 n\log\log n)$ edges with the following property: For any $F\subseteq P$,…
A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions.…
Since a plurality of processors in a distributed computer system working in parallel, to ensure the fault tolerance and stability of the network is an important issue in distributed systems. As the topology of the distributed network can be…
Recently, linear codes constructed from defining sets have been studied extensively. They may have nice parameters if the defining set is chosen properly. Let $ m >2$ be a positive integer. For an odd prime $ p $, let $ r=p^m $ and…
Let $V$ be a vector space over the finite field ${\mathbb F}_q$. A $q$-Steiner system, or an $S(t,k,V)_q$, is a collection ${\mathcal B}$ of $k$-dimensional subspaces of $V$ such that every $t$-dimensional subspace of $V$ is contained in a…
Linear codes in the projective space $\mathbb{P}_q(n)$, the set of all subspaces of the vector space $\mathbb{F}_q^n$, were first considered by Braun, Etzion and Vardy. The Grassmannian $\mathbb{G}_q(n,k)$ is the collection of all subspaces…
An anticode ${\bf C} \subset {\bf F}_q^n$ with the diameter $\delta$ is a code in ${\bf F}_q^n$ such that the distance between any two distinct codewords in ${\bf C}$ is at most $\delta$. The famous Erd\"{o}s-Kleitman bound for a binary…
We study the problem of constructing strong approximate unitary $k$-designs on $D$-dimensional grids (and more generally on Cartesian products of graphs), building on the work of Schuster et al. arXiv:2509.26310 which establishes strong…
A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice X. Our characterization…
In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive…
A construction is presented that allows to produce subspace codes of long length using subspace codes of shorter length in combination with a rank metric code. The subspace distance of the resulting code, called linkage code, is as good as…
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,…
We consider straight line drawings of a planar graph $G$ with possible edge crossings. The \emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum…
The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power…
Let ${\mathcal C}(\Omega)$ be the linear code arising from a projective system $\Omega$ of $\mathrm{PG}(V).$ Consider the point-line geometry $\Gamma=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon…
A strong clique in a graph is a clique intersecting every maximal independent set. We study the computational complexity of six algorithmic decision problems related to strong cliques in graphs and almost completely determine their…