Lattices, fundamental parallelepiped and dual of a lattice, shortest vectors, Blichfield's theorem

Lattice Definition. (Lattice) Given n linearly independent vectors b1, , bn ∈ R, the lattice generated by them · · · is defined as L(b1, b2, · · · bn) = { xibi|xi ∈ Z}. We refer to b1, · · · , bn as a basis of the lattice. Equivalently, if we define B as the m × n matrix whose columns are b1, , bn, then the lattice generated by B is · · · L(B) = L(b1, b2, · · · , bn) = {Bx|x ∈ Z}. We say that the rank of the lattice is n and its dimension is m. If n = m, the lattice is called a full-rank lattice. It is easy to see that, L is a lattice if and only if L is a discrete subgroup of (R , +). Remark. We will mostly consider full-rank lattices, as the more general case is not substantially different. Example. The lattice generated by (1, 0) and (0, 1) is Z, the lattice of all integers points (see Figure 1(a)). This basis is not unique: for example, (1, 1) and (2, 1) also generate Z (see Figure 1 (b)). Yet another basis of Z is given by (2005, 1) ; (2006, 1) . On the other hand, (1, 1) , (2, 0) is not a basis of Z: instead, it generates the lattice of all integer points whose coordinates sum to an even number (see Figure 1 (c)). All the examples so far were of full-rank lattices. An example of a lattice that is not full is L((2, 1) ) (see Figure 1(d)). It is of dimension 2 and of rank 1. Finally, the lattice Z = L((1)) is a one-dimensional full-rank lattice.