English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e 1 , ..., e n denoting the standard basis vectors of $$\sf{R}$$ n and letting x k = e 1 + ··· + e k (k = 1, 2, ..., n), the set $$ {\cal I}^n_k={\bf x}_{\bf k}^H=\{ {\bf x}_{\bf k}^g \mbox{} | \mbox{} g \in H \}$$ is the vertex set of a generalized regular hyperoctahedron in $$\sf{R}$$ n .A finite set $${\cal X} \subset \sf{R}^n$$ with a weight function $$w: {\cal X} \rightarrow \sf{R}^+$$ is called a Euclidean t-design, if $$ \sum_{r \in R} W_r \bar{f}_{S_{r}} = \sum_{{\bf x} \in {\cal X}} w({\bf x}) f({\bf x})$$ holds for every polynomial f of total degree at most t; here R is the set of norms of the points in $${\cal X}$$ ,W r is the total weight of all elements of $${\cal X}$$ with norm r, S r is the n-dimensional sphere of radius r centered at the origin, and $$\bar{f}_{S_{r}}$$ is the average of f over S r .Here we consider Euclidean designs which are supported by orbits of the hyperoctahedral group. Namely, we prove that any Euclidean design on a union of generalized hyperoctahedra has strength (maximum t for which it is a Euclidean design) equal to 3, 5, or 7.We find explicit necessary and sufficient conditions for when this strength is 5 and for when it is 7.In order to establish our classification, we translate the above definition of Euclidean designs to a single equation for t = 5, a set of three equations for t = 7, and a set of seven equations for t = 9.Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), proved a Fisher-type inequality $$|{\cal X}| \geq N(n,p,t)$$ for the minimum size of a Euclidean t-design in $$\sf{R}$$ n on p = |R| concentric spheres (assuming that the design is antipodal if t is odd).A Euclidean design with exactly N (n, p, t) points is called tight. We exhibit new examples of antipodal tight Euclidean designs, supported by orbits of the hyperoctahedral group, for N(n, p, t) = (3, 2, 5), (3, 3, 7), and (4, 2, 7).

Orbits of the hyperoctahedral group as Euclidean designs