1,198 research outputs found
Few-cosine spherical codes and Barnes-Wall lattices
Using Barnes-Wall lattices and 1-cocycles on finite groups of monomial
matrices, we give a procedure to construct tricosine spherical codes. This was
inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and
Morris discovered in studies of the universally optimal property. It has 64
vectors and cosines . We construct the {\it Optimism Code}, a
4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are
. Its automorphism group has shape .
The Optimism Code contains a subcode related to the BCGM code. The Optimism
Code implies existence of a nonlinear binary code with parameters ,
a Nordstrom-Robinson code, and gives a context for determining its automorphism
group, which has form .Comment: 24 page
Rank 72 high minimum norm lattices
Given a polarization of an even unimodular lattice and integer , we
define a family of unimodular lattices . Of special interest are
certain of rank 72. Their minimum norms lie in . Norms
4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for
rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a
. We note a few in dimensions 96, 120 and 128 with
moderately high minimum norms.Comment: submitte
Midwest cousins of Barnes-Wall lattices
Given a rational lattice and suitable set of linear transformations, we
construct a cousin lattice. Sufficient conditions are given for integrality,
evenness and unimodularity. When the input is a Barnes-Wall lattice, we get
multi-parameter series of cousins. There is a subseries consisting of
unimodular lattices which have ranks , for odd integers
and integers . Their minimum norms are
moderately high: .Comment: 33 pages; submitte
Corrections and additions to `` Pieces of : existence and uniqueness for Barnes-Wall and Ypsilanti lattices. ''
Mainly, we correct the uniqueness result by adding a projection requirement
to condition X and give a better proof for the equivalence of commutator
density, 2/4-generation and 3/4-generation.Comment: about 8 pages; submitte
Involutions on the the Barnes-Wall lattices and their fixed point sublattices, I
We study the sublattices of the rank Barnes-Wall lattices \bw d which
occur as fixed points of involutions. They have ranks (for dirty
involutions) or (for clean involutions), where , the
defect, is an integer at most . We discuss the involutions on \bw
d and determine the isometry groups of the fixed point sublattices for all
involutions of defect 1. Transitivity results for the Bolt-Room-Wall group on
isometry types of sublattices extend those in \cite{bwy}. Along the way, we
classify the orbits of on the Reed-Muller codes and
describe {\it cubi sequences} for short codewords, which give them as Boolean
sums of codimension 2 affine subspaces.Comment: 39 page
Research topics in finite groups and vertex algebras
We suggest a few projects for studying vertex algebras with emphasis on
finite group viewpoints.Comment: dedicated to Geoffrey Maso
Rank one lattice type vertex operator algebras and their automorphism groups
Let L be a positive definite even lattice of rank one and V_L^+ be the fixed
points of the lattice VOA V_L associated to L under an automorphism of V_L
lifting the -1$ isometry of L. A set of generators and the full automorphism
group of V_L^+ are determined.Comment: Latex, 15 pages, final version for publication in J. Algebr
Automorphism groups and derivation algebras of finitely generated vertex operator algebras
We investigate the general structure of the automorphism group and the Lie
algebra of derivations of a finitely generated vertex operator algebra. The
automorphism group is isomorphic to an algebraic group. Under natural
assumptions, the derivation algebra has an invariant bilinear form and the
ideal of inner derivations is nonsingular.Comment: latex 17 pages, to appear in Michigan Math.
Determinants for integral forms in lattice type vertex operator algebras
We prove a determinant formula for the standard integral form of a lattice
vertex operator algebra
A moonshine path from to the monster
One would like an explanation of the provocative McKay and Glauberman-Norton
observations connecting the extended -diagram with pairs of 2A involutions
in the Monster sporadic simple group. We propose a down-to-earth model for the
3C-case which exhibits a logic to these connections.Comment: this manuscript is the same as the 11 October 2009 arxiv version
except for a change of titl
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