# Lattice-based code

## Description

Encodes states (codewords) in coordinates of a lattice in the \(n\)-dimensional real coordinate space \(\mathbb{R}^n\). The number of codewords may be infinite because the coordinate space is infinite-dimensional, so various restricted versions have to be constructed in practice. Since lattices are closed under addition, lattice-based codes can be thought of as linear codes over the reals.

## Parent

- Error-correcting code (ECC) — Error-correcting codes are defined for a finite alphabet, so only finite lattice-based codes are children.

## Child

- Niset-Andersen-Cerf code — The Niset-Andersen-Cerf code encodes two coherent states at a time with arbitrary complex values, making it analogous to a lattice-based (classical) code encoding two points in \(\mathbb{R}^2\). The code does not encode any quantum information since superpositions of the coherent states are not stored.

## Cousins

- Linear code — Since lattices are closed under addition, lattice-based codes can be thought of as linear codes over the reals.
- Group-based code — Group-based codes whose alphabet is based on the group \(\mathbb{R}\) are lattice-based codes.
- Multi-mode GKP code — Multimode GKP codes are quantum analogues of lattice-based codes.

## Page edit log

- Victor V. Albert (2022-02-16) — most recent

## Zoo code information

## Cite as:

“Lattice-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/points_into_lattices