Related papers: Rank 72 high minimum norm lattices
We extend the results of Ozeki on the configurations of extremal even unimodular lattices. Specifically, we show that if L is such a lattice of rank 56, 72, or 96, then L is generated by its minimal-norm vectors.
The highest possible minimal norm of a unimodular lattice is determined in dimensions n <= 33. There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8*10^20 in dimension 33).…
In an earlier paper (math.NT/9906019) we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that…
It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24] +2, unless n = 23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the…
An even unimodular 72-dimensional lattice $\Gamma $ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant $-7$.…
All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in $\mathbb{Q}(\sqrt{-3})$ are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.
We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are even and unimodular \Z-lattices. Using this classification we construct the associated theta series, which are Hermitian modular forms, and…
We classify the unimodular Euclidean integral lattices of rank 29 by developing an elementary, yet very efficient, inductive method. As an application, we determine the isometry classes of even lattices of rank at most 28 and prime…
It is shown that extremal $2$-modular lattices of ranks $32$ and $48$ are generated by their vectors of minimal norm. In the proof, we use certain properties of the difference of normalized Hecke eigenforms. We refer to them as the…
We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case r=1.
In this paper, we construct odd unimodular lattices in dimensions n=36,37 having minimum norm 3 and 4s=n-16, where s is the minimum norm of the shadow. We also construct odd unimodular lattices in dimensions n=41,43,44 having minimum norm 4…
A series of integral lattices parametrised by integers $k,m,n$ are introduced and investigated, where $n$ is the rank of the lattice, including the root lattices described in a uniform way and unimodular lattices such as the Niemeier…
The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are computed using the classification of finite simple groups. Restrictions on the possible automorphisms of…
It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is…
Odd, positive-definite, integral, unimodular lattices N of rank 24 were classified by Borcherds. There are 273 isometry classes of such lattices. Associated to them are vertex superalgebras $V_N$ of central charge c=24. We show that at…
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…
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…
We show that there is a unique extremal even unimodular lattice of dimension 48 which has an automorphism of order 5 of type 5-(8,16)-8. Since the three known extremal lattices do not admit such an automorphism, this provides a new example…
We classify the unimodular equivalence classes of inclusion-minimal polygons with a certain fixed lattice width. As a corollary, we find a sharp upper bound on the number of lattice points of these minimal polygons.
The paper narrows down the possible automorphisms of extremal even unimodular lattices of dimension 72. With extensive computations in {\sc Magma} using the very sophisticated algorithm for computing class groups of algebraic number fields…