6,990 research outputs found
Spherical designs and lattices
In this article we prove that integral lattices with minimum <= 7 (or <= 9)
whose set of minimal vectors form spherical 9-designs (or 11-designs
respectively) are extremal, even and unimodular. We furthermore show that there
does not exist an integral lattice with minimum <=11 which yields a 13-design.Comment: The final publication is available at
http://link.springer.com/article/10.1007%2Fs13366-013-0155-
Spherical Designs and Heights of Euclidean Lattices
We study the connection between the theory of spherical designs and the
question of extrema of the height function of lattices. More precisely, we show
that a full-rank n-dimensional Euclidean lattice, all layers of which hold a
spherical 2-design, realises a stationary point for the height function, which
is defined as the first derivative at 0 of the spectral zeta function of the
associated flat torus. Moreover, in order to find out the lattices for which
this 2-design property holds, a strategy is described which makes use of theta
functions with spherical coefficients, viewed as elements of some space of
modular forms. Explicit computations in dimension up to 7, performed with
Pari/GP and Magma, are reported.Comment: 22 page
Conformal Designs based on Vertex Operator Algebras
We introduce the notion of a conformal design based on a vertex operator
algebra. This notation is a natural analog of the notion of block designs or
spherical designs when the elements of the design are based on self-orthogonal
binary codes or integral lattices, respectively. It is shown that the subspaces
of fixed degree of an extremal self-dual vertex operator algebra form conformal
11-, 7-, or 3-designs, generalizing similar results of Assmus-Mattson and
Venkov for extremal doubly-even codes and extremal even lattices. Other
examples are coming from group actions on vertex operator algebras, the case
studied first by Matsuo. The classification of conformal 6- and 8-designs is
investigated. Again, our results are analogous to similar results for codes and
lattices.Comment: 35 pages with 1 table, LaTe
Spherical 2-Designs and Lattices from Abelian Groups
We consider lattices generated by finite Abelian groups. The main result says that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame
Boris Venkov's Theory of Lattices and Spherical Designs
Boris Venkov passed away on November 10 2011 just 5 days before his 77th
birthday. This article gives a short survey of the mathematical work of Boris
Venkov in this direction
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