# CONFIGURATIONS OF EXTREMAL EVEN UNIMODULAR LATTICES

@article{Kominers2009CONFIGURATIONSOE,
title={CONFIGURATIONS OF EXTREMAL EVEN UNIMODULAR LATTICES},
author={Scott Duke Kominers},
journal={International Journal of Number Theory},
year={2009},
volume={05},
pages={457-464}
}
• S. Kominers
• Published 21 June 2007
• Mathematics
• International Journal of Number Theory
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.
7 Citations
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## References

SHOWING 1-10 OF 18 REFERENCES
Even unimodular Euclidean lattices of dimension 32.II
The problem of generating 32-dimensional even unimodular Euclidean lattices by vectors of a given length is studied fully. It is proved that, apart from 15 exceptions, such lattices are generated by
On even unimodular positive definite quadratic lattices of rank 32
Let Z be the ring of rational integers, and ~ the field of rational numbers. A finitely generated Z-module L in Q" with a positive definite metric is called a quadratic lattice. Since we treat only
Odd unimodular lattices of minimum 4
• Mathematics
• 2002
where n is the dimension of the lattice. The first case where this bound is not known to be tight is n = 72. It is much more difficult to obtain a good bound for the minimum of an odd unimodular
Lattices and Codes
This section introduces the basic concept of a lattice in ℝ n and describes the construction of lattices using LaSalle's inequality.
A Course in Arithmetic
Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on
Sphere Packings, Lattices and Groups
• Mathematics
Grundlehren der mathematischen Wissenschaften
• 1988
The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to