## Description

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.

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## Cousins

- Dual linear code — Dual lattices are lattice analogues of dual codes.
- Body-centered cubic (bcc) lattice code — The bcc and fcc lattices are dual to each other.

## Page edit log

- Victor V. Albert (2022-02-25) — most recent

## Cite as:

“Dual lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dual_lattice