Related papers: Bounds on equiangular lines and on related spheric…

In dimension $n=1$ we obtain $L^{p_1}(\mathbb R) \times\dots\times L^{p_m}(\mathbb R)$ to $L^p(\mathbb R)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples…

We introduce the concepts of complex Grassmannian codes and designs. Let G(m,n) denote the set of m-dimensional subspaces of C^n: then a code is a finite subset of G(m,n) in which few distances occur, while a design is a finite subset of…

We prove a lower bound of $\Omega (d^{3/2} \cdot (2/\sqrt{3})^d)$ on the kissing number in dimension $d$. This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar…

A subset $A$ of a vector space $X$ is called $\alpha$-lineable whenever $A$ contains, except for the null vector, a subspace of dimension $\alpha$. If $X$ has a topology, then $A$ is $\alpha$-spaceable if such subspace can be chosen to be…

We Use the method of linearly independent polynomials to derive an upper bound for the cardinality of a spherical s-distance set F where the sum of distinct inner products of any two elements from F is zero. Our result generalizes the…

A locally recoverable code (LRC code) is a code over a finite alphabet such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. Bounds on the rate and distance of such codes have…

This is a survey of metric properties of non-Euclidean conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the…

We prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent then they are additively equivalent.

Upper bounds on the minimum Lee distance of codes that are linear over ${\mathbb Z}_q$, $q=p^t$, $p$ prime are discussed. The bounds are Singleton like, depending on the length, rank, and alphabet size of the code. Codes meeting such bounds…

We show that the maximum cardinality of an equiangular line system in 14 and 16 dimensions is 28 and 40, respectively, thereby solving a longstanding open problem. We also improve the upper bounds on the cardinality of equiangular line…

We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost…

We consider list-decoding in the zero-rate regime for two cases: the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal $\tau \in [0,1]$ for which there exists an arrangement of $M$ balls of…

A finite set $X$ in the Euclidean unit sphere is called an $s$-distance set if the set of distances between any distinct two elements of $X$ has size $s$. We say that $t$ is the strength of $X$ if $X$ is a spherical $t$-design but not a…

We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their…

In this paper we give the enumeration formulas for Euclidean self-dual skew-cyclic codes over finite fields when $(n,|\theta|)=1$ and for some cases when $(n,|\theta|)>1,$ where $n$ is the length of the code and $|\theta|$ is the order of…

Let $X$ be an ordered vector space. The net $\{x_\alpha\}\subseteq X$ is semi unbounded order convergent to $x$ (in symbol $x_\alpha\xrightarrow{suo}x$), if there is a net $\{y_\beta\}$, possibly over a different index set, such that…

We consider the distances between a line and a set of points in the plane defined by the L^p-norms of the vector consisting of the euclidian distance between the single points and the line. We determine lines with minimal geometric…

The maximum possible number of non-overlapping unit spheres that can touch a unit sphere in $n$ dimensions is called kissing number. The problem for finding kissing numbers is closely connected to the more general problems of finding bounds…

An arc is a set of vectors of the $k$-dimensional vector space over the finite field with $q$ elements ${\mathbb F}_q$, in which every subset of size $k$ is a basis of the space, i.e. every $k$-subset is a set of linearly independent…

We improve the previously best known upper bounds on the sizes of $\theta$-spherical codes for every $\theta<\theta^*\approx 62.997^{\circ}$ at least by a factor of $0.4325$, in sufficiently high dimensions. Furthermore, for sphere packing…