A coding problem for pairs of subsets

Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap B_2=\emptyset$). Define the distance of these pairs by $d(\{A_1,A_2\} ,\{B_1,B_2\})=\min \{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\}$. This is the minimum number of elements of $A_1\cup A_2$ one has to move to obtain the other pair $\{B_1,B_2\}$. Let $C(n,k,d)$ be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs is at least $d$. Here we establish a conjecture of Brightwell and Katona concerning an asymptotic formula for $C(n,k,d)$ for $k,d$ are fixed and $n\to \infty$. Also, we find the exact value of $C(n,k,d)$ in an infinite number of cases, by using special difference sets of integers. Finally, the questions discussed above are put into a more general context and a number of coding theory type problems are proposed.Comment: 11 pages (minor changes, and new citations added

Covering by homothets and illuminating convex bodies

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number α and a convex body B, gα⁡(B) is the infimum of α-powers of finitely many homothety coefficients less than 1 such that there is a covering of B by translative homothets with these coefficients. hα⁡(B) is the minimal number of directions such that the boundary of B can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than α. In this paper, we prove that gα⁡(B)≤hα⁡(B), find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that hα⁡(B)\u3e2d−α for almost all α and d when B is the d-dimensional cube, thus disproving the conjecture from Brass, Moser, and Pach [Research problems in discrete geometry, Springer, New York, 2005]

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

On identifying codes that are robust against edge changes

AbstractAssume that G=(V, E) is an undirected graph, and C⊆V. For every v∈V, denote Ir(G; v)={u∈C: d(u,v)≤r}, where d(u,v) denotes the number of edges on any shortest path from u to v in G. If all the sets Ir(G; v) for v∈V are pairwise different, and none of them is the empty set, the code C is called r-identifying. The motivation for identifying codes comes, for instance, from finding faulty processors in multiprocessor systems or from location detection in emergency sensor networks. The underlying architecture is modelled by a graph. We study various types of identifying codes that are robust against six natural changes in the graph; known or unknown edge deletions, additions or both. Our focus is on the radius r=1. We show that in the infinite square grid the optimal density of a 1-identifying code that is robust against one unknown edge deletion is 1/2 and the optimal density of a 1-identifying code that is robust against one unknown edge addition equals 3/4 in the infinite hexagonal mesh. Moreover, although it is shown that all six problems are in general different, we prove that in the binary hypercube there are cases where five of the six problems coincide

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems

When Does a Mixture of Products Contain a Product of Mixtures?

We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable probability distributions, perfectly reconstructible inputs, Hamming modes, zonotopes and zonosets, point configurations in hyperplane arrangements, linear threshold codes, and multi-covering numbers of hypercubes. As a motivating application, we prove results on the relative representational power of mixtures of product distributions and products of mixtures of pairs of product distributions (RBMs) that formally justify widely held intuitions about distributed representations. In particular, we show that a mixture of products requiring an exponentially larger number of parameters is needed to represent the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table