Lattices: Properties

• Rank, determinant, basis, basis matrix, inner product matrix, Gram matrix, centre density, testing for integrality and evenness, index in a superlattice
• Minimum of a lattice (which can also be asserted)
• Kissing number of a lattice
• Theta series of a lattice
• Enumeration of all short vectors of a lattice having norm in a given range
• Enumeration of all shortest vectors of a lattice
• Enumeration of all vectors of a lattice having squared distance from a vector (possibly) outside the lattice in a given range
• Enumeration of all vectors of a lattice closest to a vector (possibly) outside the lattice
• Process to enumerate short or close vectors of a lattice thereby allowing manual looping over short vectors having norm in a given range or close vectors having squared distance in a given range
• Pure lattice of a lattice over $\mbox{\bf Z}$ or $\mbox{\bf Q}$
• Construction of a fundamental Voronoi cell of a small-dimensional lattice
• Holes, deep holes and covering radius of a lattice
• Successive minima of a lattice

Magma includes a highly optimized algorithm for enumerating all vectors of a lattice of a given norm. This algorithm is used for computing the minimum, the shortest vectors, short vectors in a given range, and vectors close to or closest to a given vector (possibly) outside the lattice. As an example, the 98280 (normalized) shortest vectors of the Leech lattice $\Lambda_{24}$ are constructed in 6.8 seconds. The genus of the 12-dimensional Coxeter-Todd lattice K₁₂ is enumerated in 16 seconds and has 16 classes of lattices and mass $4649359/4213820620800 \approx 0.000001103359$.