Dual lattice code 

Also called the reciprocal or polar lattice code. For any \(n\)-dimensional lattice \(L\), 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.