Let $\textrm{S}(n,t,k)$ be the maximum size of a code containing only vectors of the $k$th shell of the integer lattice $\mathbb{Z}^n$ such that the inner product between distinct vectors does …
Let $\textrm{S}(n,t,k)$ be the maximum size of a code containing only vectors of the $k$th shell of the integer lattice $\mathbb{Z}^n$ such that the inner product between distinct vectors does not exceed $t$. In this paper we compute lower bounds for $\textrm{S}(n,t,k)$ for small values of $n$, $t$ and $k$ by carrying out computer searches for codes with prescribed automorphisms. We prescribe groups of signed permutation automorphisms acting transitively on the pairs of coordinates and coordinate values as well as other closely related groups of automorphisms. Several of the constructed codes lead to improved lower bounds for spherical codes.
A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds …
A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds for coherence. In this paper two infinite families of real and complex biangular line packings are presented. New packings achieve equality in the real or complex second Levenshtein bound respectively. Both infinite families are constructed by analyzing well known representations of the finite groups SL(2,Fq). Until now the only known infinite families meeting the second Levenshtein bounds were related to the maximal sets of mutually unbiased bases (MUB). Similarly to the line packings related to the maximal sets of MUBs, the line packings presented here are related to the maximal sets of mutually unbiased weighing matrices. Another similarity is that the new packings are projective 2-designs. The latter property together with sufficiently large cardinalities of the new packings implies some improvement on largest known cardinalities of real and complex biangular tight frames.
A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds …
A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds for coherence. In this paper two two infinite families of real and complex biangular line packings are presented. New packings achieve equality in the real or complex second Levenshtein bound respectively. Both infinite families are constructed by analyzing well known representations of the finite groups $\textrm{SL}(2,\mathbb{F}_q)$. Until now the only known infinite familes meeting the second Levenshtein bounds were related to the maximal sets of mutually unbiased bases (MUB). Similarly to the line packings related to the maximal sets of MUBs, the line packings presented here are related to the maximal sets of mutually unbiased weighing matrices. Another similarity is that the new packings are projective 2-designs. The latter property together with sufficiently large cardinalities of the new packings implies some improvement on largest known cardinalities of real and complex biangular tight frames (BTF).
A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds …
A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds for coherence. In this paper two infinite families of real and complex biangular line packings are presented. New packings achieve equality in the real or complex second Levenshtein bound respectively. Both infinite families are constructed by analyzing well known representations of the finite groups SL$(2,\mathbb{F}_q)$. Until now the only known infinite familes meeting the second Levenshtein bounds were related to the maximal sets of mutually unbiased bases (MUB). Similarly to the line packings related to the maximal sets of MUBs, the line packings presented here are related to the maximal sets of mutually unbiased weighing matrices. Another similarity is that the new packings are projective 2-designs. The latter property together with sufficiently large cardinalities of the new packings implies some improvement on largest known cardinalities of real and complex biangular tight frames.
We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal …
We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.
An equiangular tight frame (ETF) is a set of equal norm vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as …
An equiangular tight frame (ETF) is a set of equal norm vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design, quantum information theory, compressed sensing, and algebraic coding theory. ETFs seem to be rare, and only a few methods of constructing them are known. In this paper, we present a new infinite family of complex ETFs that arises from hyperovals in finite projective planes. In particular, we give the first ever construction of a complex ETF of 76 vectors in a space of dimension 19. Recently, a computer-assisted approach was used to show that real ETFs of this size do not exist, resolving a longstanding open problem in this field. Our construction is a modification of a previously known technique for constructing ETFs from balanced incomplete block designs.
Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. …
Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. We prove that, if there exists a set of real units in a certain ray class field (depending on $d$) satisfying certain algebraic properties, a SIC-POVM exists, when $d$ is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at $s=0$ and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.
Article Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen. was published on January 1, 1935 in the journal Journal für die reine und angewandte Mathematik (volume 1935, issue 172).
Article Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen. was published on January 1, 1935 in the journal Journal für die reine und angewandte Mathematik (volume 1935, issue 172).
We study several interesting examples of Biangular Tight Frames (BTFs) -basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) -and examine their relationships …
We study several interesting examples of Biangular Tight Frames (BTFs) -basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) -and examine their relationships with Equiangular Tight Frames (ETFs) -basis-like systems which admit exactly one frame angle (of minimal coherence).We develop a general framework of so-called Steiner BTFs -which includes the well-known Steiner ETFs as special cases; surprisingly, the development of this framework leads to a connection with famously open problems regarding the existence of Mersenne and Fermat primes.In addition, we demonstrate an example of a smooth parametrization of 6-vector BTFs in R 3 , where the curve "passes through" an ETF; moreover, the corresponding frame angles "deform" smoothly with the parametrization, thereby answering two questions about the rigidity of BTFs.Finally, we generalize from BTFs to (chordally) biangular tight fusion frames (BTFFs) -basis-like sets of orthogonal projections admitting exactly two distinct trace inner products -and we explain how one may think of them as generalizations of BTFs.In particular, we construct an interesting example of a BTFF corresponding to 16 2-dimensional subspaces of R 4 that "Plücker embeds" into a Steiner ETF consting of 16 vectors in R 6 , which refer to as a Plücker ETF.
Abstract Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design …
Abstract Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper, we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these association schemes are nonsymmetric.
We prove that if X is a spherical t-design and s-distance set with $t\geq 2s-3$, then X has the structure of Q-polynomial association scheme of class s. Also, we describe …
We prove that if X is a spherical t-design and s-distance set with $t\geq 2s-3$, then X has the structure of Q-polynomial association scheme of class s. Also, we describe the parameters of the association scheme.
A Grassmannian frame is a collection of unit vectors which are optimally incoherent. To date, the vast majority of explicit Grassmannian frames are equiangular tight frames (ETFs). This paper surveys …
A Grassmannian frame is a collection of unit vectors which are optimally incoherent. To date, the vast majority of explicit Grassmannian frames are equiangular tight frames (ETFs). This paper surveys every known construction of ETFs and tabulates existence for sufficiently small dimensions.
Algebraic number theory relates SIC-POVMs in dimension d > 3 to those in dimension d(d − 2). We define a SIC in dimension d(d − 2) to be aligned to …
Algebraic number theory relates SIC-POVMs in dimension d > 3 to those in dimension d(d − 2). We define a SIC in dimension d(d − 2) to be aligned to a SIC in dimension d if and only if the squares of the overlap phases in dimension d appear as a subset of the overlap phases in dimension d(d − 2) in a specified way. We give 19 (mostly numerical) examples of aligned SICs. We conjecture that given any SIC in dimension d, there exists an aligned SIC in dimension d(d − 2). In all our examples, the aligned SIC has lower dimensional equiangular tight frames embedded in it. If d is odd so that a natural tensor product structure exists, we prove that the individual vectors in the aligned SIC have a very special entanglement structure, and the existence of the embedded tight frames follows as a theorem. If d − 2 is an odd prime number, we prove that a complete set of mutually unbiased bases can be obtained by reducing an aligned SIC to this dimension.
The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when …
The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when used as a uniform point-wise discretization of an ambient space. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance arising from an Euclidean embedding, including the unitary group as a special case. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: 1) the normalized volume of a metric ball and 2) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.
The kissing number of is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin [Mittelmann and Vallentin 10 [Mittelmann and Vallentin …
The kissing number of is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin [Mittelmann and Vallentin 10 [Mittelmann and Vallentin 10] H. D. Mittelmann and F. Vallentin. "High-accuracy Semidefinite Programming Bounds for Kissing Numbers." Exp. Math. 19 (2010), 175–179.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], based on the semidefinite programming bound of Bachoc and Vallentin [Bachoc and Vallentin 08 [Bachoc and Vallentin 08] C. Bachoc and F. Vallentin. "New Upper Bounds for Kissing Numbers from Semidefinite Programming." J. Am. Math. Soc. 21 (2008), 909–924.[Crossref], [Web of Science ®] , [Google Scholar]], computed the best known upper bounds for the kissing number for several values of n ⩽ 23. In this article, we exploit the symmetry present in the semidefinite programming bound to provide improved upper bounds for n = 9, …, 23.