Related papers: Complex spherical codes with three inner products
A finite set $X$ in a complex sphere is called a complex spherical $2$-code if the number of inner products between two distinct vectors in $X$ is equal to $2$. In this paper, we characterize the tight complex spherical $2$-codes by doubly…
A finite set X in the Euclidean space is called an s-inner product set if the set of the usual inner products of any two distinct points in X has size s. First, we give a special upper bound for the cardinality of an s-inner product set on…
A spherical three-distance set is a finite collection $X$ of unit vectors in $\mathbb{R}^{n}$ such that for each pair of distinct vectors has three inner product values. We use the semidefinite programming method to improve the upper bounds…
A set $C$ of unit vectors in $\mathbb{R}^d$ is called an $L$-spherical code if $x \cdot y \in L$ for any distinct $x,y$ in $C$. Spherical codes have been extensively studied since their introduction in the 1970's by Delsarte, Goethals and…
For nonnegative integers $n_2, n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least…
A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be…
We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps.…
Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This…
A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical…
A spherical $L$-code, where $L \subseteq [-1,\infty)$, consists of unit vectors in $\mathbb{R}^d$ whose pairwise inner products are contained in $L$. Determining the maximum cardinality $N_L(d)$ of an $L$-code in $\mathbb{R}^d$ is a…
A finite set X in a metric space M is called an s-distance set if the set of distances between any two distinct points of X has size s. The main problem for s-distance sets is to determine the maximum cardinality of s-distance sets for…
A finite set of distinct vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality $s$. In this paper…
We introduce the following variant of the VC-dimension. Given $S \subseteq \{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on every…
A complex spherical code is a finite subset on the unit sphere in $\mathbb{C}^d$. A fundamental problem on complex spherical codes is to find upper bounds for those with prescribed inner products. In this paper, we determine the irreducible…
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 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…
We consider the sets of dimensions for which there is an optimal sphere packing with special regularity properties (respectively, a lattice, or a periodic set with a given bound on the number of translations, or an arbitrary periodic set).…
An $L$-spherical code is a set of Euclidean unit vectors whose pairwise inner products belong to the set $L$. We show, for a fixed $\alpha,\beta>0$, that the size of any $[-1,-\beta]\cup\{\alpha\}$-spherical code is at most linear in the…
Given an open set $T\subset [-1,1)$, we introduce the concepts of $T$-avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set $T$. We show that certain codes found in the minimal vectors of the…
A code $C \subseteq \{0, 1, 2\}^n$ of length $n$ is called trifferent if for any three distinct elements of $C$ there exists a coordinate in which they all differ. By $T(n)$ we denote the maximum cardinality of trifferent codes with length.…