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 $-3/7, -1/7, 1/7$. We construct the {\it Optimism Code}, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are $0, 1/4, -1/4, -1$. Its automorphism group has shape $2^{1+8}{\cdot}GL(4,2)$. The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters $(16,256,6)$, a Nordstrom-Robinson code, and gives a context for determining its automorphism group, which has form $2^4{:}Alt_7$.Comment: 24 page

Rank 72 high minimum norm lattices

Given a polarization of an even unimodular lattice and integer $k\ge 1$, we define a family of unimodular lattices $L(M,N,k)$. Of special interest are certain $L(M,N,3)$ of rank 72. Their minimum norms lie in $\{4, 6, 8\}$. 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 $L(M,N,3)$. We note a few $L(M,N,k)$ 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 $2^{d-1}\pm 2^{d-k-1}$, for odd integers $d\ge 3$ and integers $k=1,2, ..., \frac {d-1}2$. Their minimum norms are moderately high: $2^{\lfloor \frac d2 \rfloor -1}$.Comment: 33 pages; submitte

Corrections and additions to `` Pieces of 2d2^d: 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 $2^d$ Barnes-Wall lattices \bw d which occur as fixed points of involutions. They have ranks $2^{d-1}$ (for dirty involutions) or $2^{d-1}\pm 2^{k-1}$ (for clean involutions), where $k$, the defect, is an integer at most $\frac d 2$. 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 $AGL(d,2)$ on the Reed-Muller codes $RM(2,d)$ 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 E8E_8 to the monster

One would like an explanation of the provocative McKay and Glauberman-Norton observations connecting the extended $E_8$-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