duallattice

The dual lattice, sometimes called a duallattice, is a lattice L* associated with a lattice L in Euclidean space. It consists of all vectors y in the ambient space such that the inner product with every vector x in L is an integer: L* = { y | ⟨y, x⟩ ∈ Z for all x ∈ L }. The dual captures how L interacts with the integer-valued linear forms on the ambient space.

If L is full rank and generated by a basis with columns forming an invertible matrix B, then L = B Z^n and the dual lattice is L* = (B^{-1})^T Z^n. Equivalently, L* contains all y for which B^T y ∈ Z^n. A key consequence is that the volumes satisfy vol(L) · vol(L*) = 1, so det(L) det(L*) = 1. When L has det(L) = 1 or is unimodular, L* = L.

Examples help clarify the construction. For the standard integer lattice Z^n, the dual lattice is itself: Z^n* = Z^n. If L is generated by the basis with matrix B = diag(1, 2) in R^2, then L = {(m, 2n) : m,n ∈ Z} and L* is generated by the columns of (B^{-1})^T = diag(1, 1/2), so L* = {(a, b/2) : a,b ∈ Z}.

Dual lattices play a central role in lattice-based cryptography, coding theory, and the analysis of lattices via Poisson summation. They interact with concepts such as unimodularity, covering and packing radii, and lattice reduction techniques. In many contexts, L ⊆ L*, with equality characterizing unimodular (or integral) lattices.