MAGMA Computational Algebra System

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Lattices

New Features:

• Functions that compute HKZ-reduced bases of matrices, Gram matrices and lattices have been implemented. A Hermite-Korkine-Zolotarev reduced basis starts with a shortest non-zero lattice vector, and orthogonally to the first basis vector the remaining vectors are themselves HKZ-reduced. HKZ-reduction is a very strong notion of reduction, providing bases of much better quality than LLL-reduction. It is however much more expensive to obtain. HKZ-reducing a lattice may allow the user to solve problems on a given lattice more easily, enumerating short and close vectors being two natural examples. The functions GaussReduce and GaussReduceGram are restrictions of the HKZ functions in dimension 2.
• A new SetVerbose(``HKZ'', b) flag allows the user to obtain information during the computation of an HKZ-reduced basis.
• The function EnumerationCostArray provides a priori information on the efficiency of the executions of the functions Minimum, CentreDensity, CenterDensity, KissingNumber, ShortVectors, ShortVectorsMatrix, ShortestVectors, ShortestVectorsMatrix and ThetaSeries. The information is more precise than that provided by the function EnumerationCost provided in the previous release, as EnumerationCostArray(L, u) gives a heuristic evaluation of the size of each layer of the tree to be visited during the enumeration of vectors within the prescribed norm u, rather than the sum of the sizes of the layers.
• A new Prune option has been added for the functions Minimum, CentreDensity, CenterDensity, EnumerationCost, KissingNumber, ShortVectors, ShortVectorsMatrix, ShortestVectors,
  ShortestVectorsMatrix and ThetaSeries. The Prune option is also available for the new functions EnumerationCostArray and HKZ. It allows the user to finely prune the tree to be considered during the enumeration. The output may not be correct anymore, but by using the EnumerationCostArray function, the user can heuristically estimate the running-time speed-up and the likeliness of an incorrect output.
• The function ReconstructLatticeBasis takes as input an arbitrary basis of a lattice and a full rank set of short linearly independent vectors. It returns a basis of the lattice that is not much longer than the full-rank set of linearly independent vectors.
• An improved algorithm for computing the automorphism group of an integral lattice has been developed. The algorithm can handle lattices having a much larger number of vectors of minimal norm than its predecessor. The result is that it is much faster than the old algorithm and can handle significantly larger lattices. For instance, it is able to compute the automorphism group of some of the easier lattices of dimension 48 in the Sloane-Nebe database. A similar algorithm for determining isometry of a pair of lattices is also provided.
• A new version of the lattice database, with slightly different functionality, is now available. The main feature is the addition and checking of many more automorphism groups, and similarly with $\Theta$-series. A few new lattices have been added, and some duplicates have been removed. The information about Hermitian bases has not been included, but can be added if users request the Magma group to do so.

Bug Fixes:

• Two local solubility glitches in dimension 4 for IsotropicSubspace have been fixed.

• Another local solubility problem was also fixed, and a failure to minimize in some cases (particularly dimension 6) were also fixed.

• A bug with the 2-adic genus of a lattice was fixed.