Journal of Algebraic Combinatorics
, Volume 25, Issue 4, pp 375397
First online:
Orbits of the hyperoctahedral group as Euclidean designs
 Béla BajnokAffiliated withGettysburg College Email author
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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 Fishertype inequality \({\cal X} \geq N(n,p,t)\) for the minimum size of a Euclidean tdesign 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).
Keywords
Euclidean design Spherical design Tight design Harmonic polynomial Hyperoctahedral group Title
 Orbits of the hyperoctahedral group as Euclidean designs
 Journal

Journal of Algebraic Combinatorics
Volume 25, Issue 4 , pp 375397
 Cover Date
 200706
 DOI
 10.1007/s1080100600423
 Print ISSN
 09259899
 Online ISSN
 15729192
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Euclidean design
 Spherical design
 Tight design
 Harmonic polynomial
 Hyperoctahedral group
 Authors

 Béla Bajnok ^{(1)}
 Author Affiliations

 1. Gettysburg College, Gettysburg, Pennsylvania, USA