Related papers: Large Sets with Multiplicity
In this paper, we list several interesting structures of cyclotomic polynomials: specifically relations among blocks obtained by suitable partition of cyclotomic polynomials. We present explicit and self-contained proof for all of them,…
Multi-component aggregates are being intensively researched in various fields because of their highly tunable properties and wide applications. Due to the complex configurational space of these systems, research would greatly benefit from a…
In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. (*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a…
The problem of ranking can be described as follows. We have a set of combinatorial objects $S$, such as, say, the k-subsets of n things, and we can imagine that they have been arranged in some list, say lexicographically, and we want to…
Recent work has uncovered a striking phenomenon in large-capacity neural networks: they contain blocks of contiguous hidden layers with highly similar representations. This block structure has two seemingly contradictory properties: on the…
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…
Stochastic Block Models (SBMs) are a popular approach to modeling single real-world graphs. The key idea of SBMs is to partition the vertices of the graph into blocks with similar edge densities within, as well as between different blocks.…
Many code families such as low-density parity-check codes, fractional repetition codes, batch codes and private information retrieval codes with low storage overhead rely on the use of combinatorial block designs or derivatives thereof. In…
This paper aims to generalize and unify classical criteria for comparisons of balanced lattice designs, including fractional factorial designs, supersaturated designs and uniform designs. We present a general majorization framework for…
Coding theory and combinatorial $t$-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some…
Discovering and characterizing the large-scale topological features in empirical networks are crucial steps in understanding how complex systems function. However, most existing methods used to obtain the modular structure of networks…
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…
We introduce and investigate binary $(k,k)$-designs -- combinatorial structures which are related to binary orthogonal arrays. We derive general linear programming bound and propose as a consequence a universal bound on the minimum possible…
In this paper, we propose to enumerate all different configurations belonging to a specific class of fractals: A binary initial tile is selected and a finite recursive tiling process is engaged to produce auto-similar binary patterns. For…
Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in simple situations. This paper concerns binary radix systems, i.e., positional number systems with digits…
Polymer materials have the characteristic feature that they are multiscale systems by definition. Already the description of a single molecules involves a multitude of different scales, and cooperative processes in polymer assemblies are…
Symmetry breaking for graphs and other combinatorial objects is notoriously hard. On the one hand, complete symmetry breaks are exponential in size. On the other hand, current, state-of-the-art, partial symmetry breaks are often considered…
This article discusses some difficulties in the implementation of combinatorial algorithms associated with the choice of all elements with certain properties among the elements of a set with great cardinality.The problem has been resolved…
A Kirkman triple system of order $v$, KTS$(v)$, is a resolvable Steiner triple system on $v$ elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS$(v)$ which contain as a subdesign a…
We study random composite structures considered up to symmetry that are sampled according to weights on the inner and outer structures. This model may be viewed as an unlabelled version of Gibbs partitions and encompasses multisets of…