\(\mathbb{Z}^n\) hypercubic lattice code
Description
Lattice-based code consisting of all integer vectors in \(n\) dimensions. It's generator matrix is the \(n\)-dimensional identity matrix.
Protection
The \(\mathbb{Z}\) integer lattice solves the lattice quantization problem in one dimension with a second moment of \(G_1 = 1/12\).
Parent
Cousins
- Dual lattice code — The hypercubic lattice code is odd and unimodular.
- Barnes-Wall (BW) lattice code — The hypercubic lattice is the \(m=1\) BW lattice.
- Lattice-based code — The generator matrix of a lattice-based code serves as a linear transformation that can be applied to the hypercubic lattice to obtain said code [1; Ch. 10].
- Square-lattice GKP code — GKP codewords, when written in terms of coherent states, form a square lattice in phase space.
References
- [1]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
Page edit log
- Victor V. Albert (2022-12-12) — most recent
Cite as:
“\(\mathbb{Z}^n\) hypercubic lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hypercubic