A problem on distance matrices of subsets of the Hamming cube

1986

We provide evidence for the following conjecture: a minimal covering of the binary Hamming space F~ + 2 by spheres of radius t + 1 has at most the same cardinality as a minimal covering of F~ by spheres of radius t. 1. Introduction For a vector x e F~, we denote by B(x ,t) the Hamming sphere of radius t centered at x, i.e. B(x,t) = {y e F~, d(x,y) ~ t}, where d(x ,y) is the Hamming distance between x and y. We say that C is a t-covering of F~ if C+B(O,t) =F~, where, for X , Y C F~, X + Y = {x + y , x eX, y eY}. Finally, let K(n,t) be the minimal cardinality of such a covering. If moreover C is a linear subspace of F~ (a linear code), we call it a linear t-covering, and denote by k[n,t] the minimal dimension of such a covering. We shall discuss the following conjectures.

Discrete Mathematics, 1981

2010

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree (T , d) is a metric space such that between any two of its points there is an unique arc that is isometric to an interval in R. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images x0 = π((x1 + . . . + xn)/n), where π is a contractive retraction from the ambient Banach space X onto T (such a π always exists), in order to understand the "metric" barycenter of a family of points x1, . . . , xn in a tree T . Further, we consider the metric properties of trees (such as their type and cotype). We identify various measures of compactness of metric trees (their covering numbers, ǫ-entropy and Kolmogorov widths) and the connections between them. Additionally, we prove that the limit of the sequence of the Kolmogorov widths of a metric tree is equal to its ball measure of non-compactness.

SIAM Journal on Discrete Mathematics, 2008

The edge-bandwidth B (G) of a graph G is the bandwidth of the line graph of G. More specifically, for any bijection f : E(G) → {1, 2, . . . , |E(G)|}, let B (f, G) = max{|f (e 1 )−f (e 2 )| : e 1 and e 2 are incident edges of G}, and let B (G) = min f B (f, G). We determine asymptotically the edge-bandwidth of d-dimensional grids P d n and of the Hamming graph K d n , the d-fold Cartesian product of K n . Our results are as follows.

1985

A coloring of the vertices of a graph G is a distance k coloring of G if and only if any two vertices lying on a path of length less than or equal to k are given dierent colors. Hamming graphs are Cartesian (or box) products of complete graphs. In this paper, we will consider the interaction between coding theory and distance k colorings of Hamming graphs.

On the general distance problem in $R^k$, 2022

Using the method of compression we obtain a generalized lower bound for the number of $d$-unit distances that can be formed from a set of $n$ points in the euclidean space $\mathbb{R}^k$. By letting $\mathcal{D}_{n,d}$ denotes the number of $d$-unit distances that can be formed from a set of $n$ points in $\mathbb{R}^k$, then we obtain the lower bound \begin{align} \mathcal{D}_{n,d}\gg \frac{n\sqrt{k}}{d}.\nonumber \end{align}

arXiv (Cornell University), 2021

Metric Ramsey theory is concerned with finding large well-structured subsets of more complex metric spaces. For finite metric spaces this problem was first studies by Bourgain, Figiel and Milman [21], and studied further in depth by Bartal et. al [10]. In this paper we provide deterministic constructions for this problem via a novel notion of metric Ramsey decomposition. This method yields several more applications, reflecting on some basic results in metric embedding theory. The applications include various results in metric Ramsey theory including the first deterministic construction yielding Ramsey theorems with tight bounds, a well as stronger theorems and properties, implying appropriate distance oracle applications. In addition, this decomposition provides the first deterministic Bourgain-type embedding of finite metric spaces into Euclidean space, and an optimal multi-embedding into ultrametrics, thus improving its applications in approximation and online algorithms. The decomposition presented here, the techniques and its consequences have already been used in recent research in the field of metric embedding for various applications. * This is paper is still in stages of preparation, this version is not intended for distribution. A preliminary version of this article was written by the author in 2006, and was presented in the 2007 ICMS Workshop on Geometry and Algorithms [14]. The basic result on constructive metric Ramsey decomposition and metric Ramsey theorem has also appeared in the author's lectures notes, e.g. [15].

1996

Abstract Given a subset X of vertices of the n-cube (ie, the n-dimensional Hamming space), we are interested in the solution of the traveling salesman problem; namely, the minimal length of a cycle passing through all vertices of X. For a given number M, we estimate the maximum of these lengths when X ranges over all possible choices of sets of M vertices. Asymptotically, our estimates show that for a number M of vertices growing exponentially in n, the maximum is attained for a code with maximal possible minimum distance

We consider metrics determined by hierarchical posets and give explicit formulae for the main parameters of a linear code (packing, covering and Chebyshev radii and minimal distance), in terms of the corresponding Hamming parameters. We also give ten characterizations of hierarchical poset metrics, including new characterizations and simple new proofs to the known ones.