Description
Lattice-based code consisting of all integer vectors in \(n\) dimensions. Its generator matrix is the \(n\)-dimensional identity matrix. Its automorphism group consists of all coordinate permutations and sign changes.
Protection
The \(\mathbb{Z}\) integer lattice solves the lattice quantization problem in one dimension with a second moment of \(G_1 = 1/12\). The lattice has determinant 1, kissing number \(2n\), packing radius \(1/2\), covering radius \(\sqrt{n}/2\), and density \(V_{n}/\sqrt{2^{n}(n+1)}\) (with \(V_n\) the volume of the unit \(n\)-sphere).
Parents
- Root lattice code
- Unimodular lattice code — The hypercubic lattice code is odd and unimodular.
Cousins
- 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].
- Biorthogonal spherical code — Biorthogonal spherical codewords form the minimal shell of the \(\mathbb{Z}^n\) hypercubic lattice.
- Hypercube code — Hypercube codewords form the minimal lattice shell code of the \(\mathbb{Z}^n\) hypercubic lattice when the lattice is shifted such that the center of a hypercube is at the origin.
- Square-lattice GKP code — GKP codewords, when written in terms of coherent states, form a square lattice in phase space.
- Square-octagon (4.8.8) color code — The 4.8.8 (square-octagon) tiling is obtained by applying a fattening procedure to the honeycomb lattice [2].
References
- [1]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [2]
- H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
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