arXiv subjects
Publications and source records attributed to Gabriele Nebe.
It is shown that under some additional assumption two diagonalizable integral matrices X and Y with only rational eigenvalues are conjugate in GL(n,Z) if and only if they are conjugate over all localizations. This is used to prove that for a prime p == 3 (mod 4) the adjacency matrices of the Paley graph and the Peisert graph on p^2 vertices are conjugate in GL(p^2,Z), answering a question by Peter Sin.
We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice $Λ_{16}$, the extremal 5-modular lattice $N_{16}$, the odd Barnes-Wall lattice $O_{16}$ and its dual, and one pair of new lattices $Γ_{16}$ and its dual. The latter pair belongs to a new infinite series of dual strongly perfect lattices, the sandwiched Barnes-Wall lattices, described by the authors in a previous paper. An updated table of all known strongly perfect lattices up to dimension 26 is available in the catalogue of lattices.
We study the behaviour of the minimal slope of Euclidean lattices under tensor product. A general conjecture predicts that $μ_{min}(L \otimes M) = μ_{min}(L)μ_{min}(M)$ for all Euclidean lattices $L$ and $M$. We prove that this is the case under the additional assumptions that $L$ and $M$ are acted on multiplicity-free by their automorphism group, such that one of them has at most $2$ irreducible components.
We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of (V,q). This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in (V,q).
In this paper we describe a parallel Gaussian elimination algorithm for matrices with entries in a finite field. Unlike previous approaches, our algorithm subdivides a very large input matrix into smaller submatrices by subdividing both rows and columns into roughly square blocks sized so that computing with individual blocks on individual processors provides adequate concurrency. The algorithm also returns the transformation matrix, which encodes the row operations used. We go to some lengths to avoid storing any unnecessary data as we keep track of the row operations, such as block columns of the transformation matrix known to be zero. The algorithm is accompanied by a concurrency analysis which shows that the improvement in concurrency is of the same order of magnitude as the number of blocks. An implementation of the algorithm has been tested on matrices as large as $1 000 000\times 1 000 000$ over small finite fields.
New series of $2^{2m}$-dimensional universally strongly perfect lattices $Λ_I $ and $Γ_J $ are constructed with $$2BW_{2m} ^{\#} \subseteq Γ_J \subseteq BW_{2m} \subseteq Λ_I \subseteq BW _{2m}^{\#} .$$ The lattices are found by restricting the spin representations of the automorphism group of the Barnes-Wall lattice to its subgroup ${\mathcal U}_m:={\mathcal C}_m (4^H_{\bf 1}) $. The group ${\mathcal U}_m$ is the Clifford-Weil group associated to the Hermitian self-dual codes over ${\bf F} _4$ containing ${\bf 1}$, so the ring of polynomial invariants of ${\mathcal U}_m$ is spanned by the genus-$m$ complete weight enumerators of such codes. This allows us to show that all the ${\mathcal U}_m$ invariant lattices are universally strongly perfect. We introduce a new construction, $D^{(cyc)}$ for chains of (extended) cyclic codes to obtain (bounds on) the minimum of the new lattices.
We develop a network coding technique based on flags of subspaces and a corresponding network channel model. To define error correcting codes we introduce a new distance on the flag variety, the Grassmann distance on flags and compare it to the commonly used gallery distance for full flags.
We classify all one-class genera of admissible lattice chains of length at least 2 in hermitian spaces over number fields.
The rational invariants of the SL_2(q)-invariant quadratic forms on the real irreducible representations are determined. There is still one open question (see Remark 6.5) if q is an even square.
We design an algorithm to find certain partial permutation representations of a finitely presented group $G$ (the bricks) that may be combined to a transitive permutation representation of $G$ (the mosaic) on the disjoint union.
Let K be a cyclic Galois extension of degree f over k and T a generator of the Galois group. For any v=(v_1,... , v_m)\in K^m such that v is linearly independent over k, and any 0< d < m the Gabidulin-like code C(v, T , d) is a maximum rank distance code in the space of f times m matrices over k of dimension fd. This construction unifies the ones available in the literature. We characterise the K-linear codes that are Gabidulin-like codes and determine their rank-metric automorphism group.
We determine the automorphism group of Gabidulin codes of full length and characterise when these codes are equivalent to self-dual codes.
A classical result of Conway and Pless is that a natural projection of the fixed code of an automorphism of odd prime order of a self-dual binary linear code is self-dual. In this paper we prove that the same holds for involutions under some (quite strong) conditions on the codes. In order to prove it, we introduce a new family of binary codes: the semi self-dual codes. A binary self-orthogonal code is called semi self-dual if it contains the all-ones vector and is of codimension 2 in its dual code. We prove upper bounds on the dual distance of semi self-dual codes. As an application we get the following: let C be an extremal self-dual binary linear code of length 24m and s in Aut(C) be a fixed point free automorphism of order 2. If m is odd or if m=2k with binom{5k-1}{k-1} odd then C is a free F_2 -module. This result has quite strong consequences on the structure of the automorphism group of such codes.
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 written by Steve Donnelly it is shown that the extremal even unimodular lattice $Γ_{72}$ from \cite{dim72} is the unique extremal even unimodular lattice of dimension 72 that admits a large automorphism, where a $d\times d$ matrix is called large, if its minimal polynomial has an irreducible factor of degree $> d/2$.
We describe an algorithm, meant to be very general, to compute a presentation of the group of units of an order in a (semi)simple algebra over Q. Our method is based on a generalisation of Voronoï's algorithm for computing perfect forms, combined with Bass-Serre theory. It differs essentially from previously known methods to deal with such questions, e.g. for units in quaternion algebras. We illustrate this new algorithm by a series of examples where the computations are carried out completely.
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 of an extremal even unimodular lattice in dimension 48.
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 computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72, whose automorphism group contains the symmetric group of degree 3, the alternating group of degree 4 or the dihedral group of order 8. Combining this with the known results in the literature one obtains that Aut(C) has order at most 5 or isomorphic to the elementary abelian group of order 8.