A finite set $X \seq \RR^n$ with a weight function $w : X \longrightarrow \RR_{>0}$ is called \emph{Euclidean $t$-design} in $\RR^n$ (supported by $p$ concentric spheres) if the following condition holds: \[ \sum_{i=1}^p \frac{w(X_i)}{|S_i|}\int_{S_i} f(\boldsymbol x)dσ_i(\boldsymbol x) =\sum_{\boldsymbol x \in X}w(\boldsymbol x) f(\boldsymbol x), \] for any polynomial $f(\boldsymbol x) \in \mbox{Pol}(\RR^n)$ of degree at most $t$. Here $S_i \seq \RR^n$ is a sphere of radius $r_i \geq 0,$ $X_i=X \cap S_i,$ and $σ_i(\boldsymbol x)$ is an $O(n)$-invariant measure on $S_i$ such that $|S_i|=r_i^{n-1}|S^{n-1}|$, with $|S_i|$ is the surface area of $S_i$ and $|S^{n-1}|$ is a surface area of the unit sphere in $\RR^n$.
Recently, Bajnok (2006) constructed tight Euclidean $t$-designs in the plane ($n=2$) for arbitrary $t$ and $p.$ In this paper we show that for case $t=6$ and $p=2,$ tight Euclidean $6$-designs constructed by Bajnok is the unique configuration in $\RR^n$, for $2 \leq n \leq 8.$
This paper supplies additions to our paper in Linear Algebra Appl. 510 (2016) 395--420 on integral spans of tight frames in Euclidean spaces. In that previous paper, we considered the case of an equiangular tight frame (ETF), proving that if its integral span is a lattice then the frame must be rational, but overlooking a simple argument in the reverse direction. Thus our first result here is that the integral span of an ETF is a lattice if and only if the frame is rational. Further, we discuss conditions under which such lattices are eutactic and perfect and, consequently, are local maxima of…
I give an elementary proof that a quasi-symmetric design without repeated blocks on $v$ points has at most $v\choose2$ blocks, with equality if and only if it is a tight $4$-design.
We complete the classification of tight $4$-designs in Hamming association schemes $H(n,q)$, i.e., that of tight orthogonal arrays of strength $4$, which had been open since a result by Noda (1979). To do so, we construct an association scheme attached to a tight $4$-design in $H(n,q)$ and analyze its triple intersection numbers to conclude the non-existence in all open cases.
Given a finite subset of a sphere or projective space, known as a design, we can compute the strength and angle set of that design. When the strength and angle set meet certain bounds, the design is called tight. Hoggar sought to prove that, aside from certain known cases, the angle sets of tight projective designs must be rational. Lyubich found a counter-example and provided a repair for Hoggar's proof but excluded the exceptional octonion projective cases. This note extends Lyubich's repair of Hoggar's proof to the remaining projective cases and extends the proof to all spherical cases. It …
The two strictly projective tight 5-designs are the lines spanned by the short vectors of the Leech lattice and a set of points in the octonion projective plane that define a generalized hexagon of order (2,8). A previous paper introduced a common construction that can generate these two tight 5-designs. This paper describes the same construction in terms of octonion arithmetic. An octonion integer construction of the Leech lattice is described using properties of the octonion integers taken modulo 2. The Leech lattice automorphism group is constructed from octonion reflections. The common con…
In addition to the vertices of the regular hexagon and icosahedron, there are precisely two strictly projective tight 5-designs: one constructed from the short vectors of the Leech lattice and the other corresponding to a generalized hexagon structure in the octonion projective plane. This paper describes a new connection between these two strictly projective tight 5-designs -- a common construction using octonions. Certain octonion involutionary matrices act on a three-dimensional octonion vector space to produce the first 5-design and these same matrices act on the octonion projective plane …
We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are …
An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs. The …
