The extremal 3-modular lattice $[\pm G_2(3)]_{14}$ with automorphism group $C_2 \times G_2(\F_3) ... more The extremal 3-modular lattice $[\pm G_2(3)]_{14}$ with automorphism group $C_2 \times G_2(\F_3) $ is the unique dual strongly perfect lattice of dimension 14.
We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are ... more 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 compute the filtration of cusp forms.
ABSTRACT We classify the lattices of rank 16 over the Eisenstein integers which are even unimodul... more ABSTRACT We classify the lattices of rank 16 over the Eisenstein integers which are even unimodular -lattices (of dimension 32). There are exactly 80 unitary isometry classes.
We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finit... more We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finite subgroups of the unit group of a maximal order in some simple $\QQ $-algebra. This may be used to show in small cases that non-conjugate orders have non-isomorphic unit groups.
A certain family of orthogonal groups (called "Clifford groups" by G. E. Wall) has arisen in a va... more A certain family of orthogonal groups (called "Clifford groups" by G. E. Wall) has arisen in a variety of different contexts in recent years. These groups have a simple definition as the automorphism groups of certain generalized Barnes-Wall lattices. This leads to an especially simple construction for the usual Barnes-Wall lattices. This is based on the third author's talk at the Forney-Fest, M.I.T., March 2000, which in turn is based on our paper "The Invariants of the Clifford Groups", Designs, Codes, Crypt., 24 (2001), 99--121, to which the reader is referred for further details and proofs.
The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weigh... more The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules. The proof of the theorem uses a categorical approach, and will be the subject of a forthcoming book. However, the theorem can be stated and applied without using category theory, and we illustrate it here by applying it to generalized doubly-even codes over fields of characteristic 2, doubly-even codes over the integers modulo a power of 2, and self-dual codes over the noncommutative ring $\F_q + \F_q u$, where $u^2 = 0$..
The extremal 3-modular lattice $[\pm G_2(3)]_{14}$ with automorphism group $C_2 \times G_2(\F_3) ... more The extremal 3-modular lattice $[\pm G_2(3)]_{14}$ with automorphism group $C_2 \times G_2(\F_3) $ is the unique dual strongly perfect lattice of dimension 14.
We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are ... more 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 compute the filtration of cusp forms.
ABSTRACT We classify the lattices of rank 16 over the Eisenstein integers which are even unimodul... more ABSTRACT We classify the lattices of rank 16 over the Eisenstein integers which are even unimodular -lattices (of dimension 32). There are exactly 80 unitary isometry classes.
We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finit... more We apply Voronoi's algorithm to compute representatives of the conjugacy classes of maximal finite subgroups of the unit group of a maximal order in some simple $\QQ $-algebra. This may be used to show in small cases that non-conjugate orders have non-isomorphic unit groups.
A certain family of orthogonal groups (called "Clifford groups" by G. E. Wall) has arisen in a va... more A certain family of orthogonal groups (called "Clifford groups" by G. E. Wall) has arisen in a variety of different contexts in recent years. These groups have a simple definition as the automorphism groups of certain generalized Barnes-Wall lattices. This leads to an especially simple construction for the usual Barnes-Wall lattices. This is based on the third author's talk at the Forney-Fest, M.I.T., March 2000, which in turn is based on our paper "The Invariants of the Clifford Groups", Designs, Codes, Crypt., 24 (2001), 99--121, to which the reader is referred for further details and proofs.
The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weigh... more The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules. The proof of the theorem uses a categorical approach, and will be the subject of a forthcoming book. However, the theorem can be stated and applied without using category theory, and we illustrate it here by applying it to generalized doubly-even codes over fields of characteristic 2, doubly-even codes over the integers modulo a power of 2, and self-dual codes over the noncommutative ring $\F_q + \F_q u$, where $u^2 = 0$..
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Papers by Gabriele Nebe