Related papers: The completion of optimal $(3,4)$-packings

In this paper the 3-way intersection problem for $S(2,4,v)$ designs is investigated. Let $b_{v}=\frac {v(v-1)}{12}$ and $I_{3}[v]=\{0,1,...,b_{v}\}\setminus\{b_{v}-7,b_{v}-6,b_{v}-5,b_{v}-4,b_{v}-3,b_{v}-2,b_{v}-1\}$. Let $J_{3}[v]=\{k|$…

We obtain a new lower bound on the largest Sidon subset of an arbitrary finite set of integers. If $H(n)$ denotes the minimum, over all $n$-element subsets of $\mathbb Z$, of the largest Sidon subset they contain, we prove that $H(n)…

In this paper, we are interested in minimizing the sum of block sizes in a pairwise balanced design, where there are some constraints on the size of one block or the size of the largest block. For every positive integers n;m, where m ? n,…

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest…

An error model with asymmetric single magnitude four error is considered. This paper is about constructions of codes correcting single error over $\mathbb{Z}_{2^{a}3^{b}r}$. Firstly, we reduce the construction of a maximal size…

We establish the existence of optimal maximal entanglement entanglement-assisted quantum $[[n,k,d;n-k]]_2$ codes for $(n,k,d)=(14,6,7)$, $(15,7,7)$, $(17,6,9)$, $(17,7,8)$, $(19,7,9)$ and $(20,7,10)$. These codes are obtained from…

Given a finite family of squares in the plane, the packing problem asks for the maximum number $\nu$ of pairwise disjoint squares among them, while the hitting problem for the minimum number $\tau$ of points hitting all of them. Clearly,…

Define a building blocks set to be a collection of n cubes (each with six sides) where each side is assigned one letter and one color from a palette of m colors. We propose a novel problem of assigning letters and colors to each face so as…

Consider a set $X\subseteq \mathbb{R}^d$ which is 1-dense, namely, it intersects every unit ball. We show that we can get from any point to any other point in $\mathbb{R}^d$ in $n$ steps so that the intermediate points are in $X$, and the…

Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show…

Network caching asks how to place contents in distributed caches so that future requests are served close to their users. Ganian, Mc Inerney and Tsigkari recently initiated the parameterized-complexity study of the problem and, for the…

We study a class of combinatorial designs called Kirkman systems, and we show that infinitely many Kirkman systems are well-distributed in a precise sense. Steiner triple systems of order $n$ can achieve a minimum block sum of $n$. Kirkman…

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least $k$ vertices is considered long. When $k \le 3$, the…

The homogeneous metric can be viewed as a natural extension of the Hamming metric to finite chain rings. It distinguishes between three types of elements: zero, non-zero elements in the socle, and elements outside the socle. Since the…

Permutation codes of length $n$ and distance $d$ is a set of permutations on $n$ symbols, where the distance between any two elements in the set is at least $d$. Subgroup permutation codes are permutation codes with the property that the…

A conforming partition of a rectilinear n-gon P (possibly with holes) is a partition of P into rectangles without using Steiner points (i.e., all corners of all rectangles must lie on the boundary of P). The stabbing number of such a…

An ordered matching of size $n$ is a graph on a linearly ordered vertex set $V$, $|V|=2n$, consisting of $n$ pairwise disjoint edges. There are three different ordered matchings of size two on $V=\{1,2,3,4\}$: an alignment…

In the (1-dimensional) bin packing problem, we are asked to pack all the given items into bins, each of capacity one, so that the number of non-empty bins is minimized. Zhu~[Chaos, Solitons \& Fractals 2016] proposed an approximation…

We consider tilings and packings of $\RR^d$ by integral translates of cubes $[0,2[^d$, which are $4\ZZ^d$-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension $d\leq…

We continue the study of Adin, Alon and Roichman [arXiv:2502.14398, 2025] on the number of steps required to sort $n$ labelled points on a circle by transpositions. Imagine that the vertices of a cycle of length $n$ are labelled by the…