Dual lattice code 

For any dimensional lattice \(L\), the dual lattice is the set of vectors whose inner products with the elements of \(L\) are integers.

More technically, the dual lattice is \begin{align} L^{\perp} = \{ y\in \mathbb{R}^{n} ~|~ x \cdot y \in \mathbb{Z} ~\forall~ x \in L\}, \tag*{(1)}\end{align} where the Euclidean inner product is used.

A lattice that is contained in its dual, \(L \subseteq L^\perp\), is called integral. The Gram matrix of such a lattice has integer entries, and its dual is contained in a suitably scaled version of itself, \(L^{\perp} \subseteq L/\det L\). Integral lattices are classified into even or odd, where the norm squared of every lattice vector is an even or odd integer, respectively.

A lattice that is equal to its dual, \(L^\perp = L\), is called unimodular or self-dual.