\(\mathbb{Z}^n\) hypercubic lattice code 

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

Cousins

References

[1]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
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Zoo Code ID: hypercubic

Cite as:
\(\mathbb{Z}^n\) hypercubic lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hypercubic
BibTeX:
@incollection{eczoo_hypercubic, title={\(\mathbb{Z}^n\) hypercubic lattice code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hypercubic} }
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Cite as:

\(\mathbb{Z}^n\) hypercubic lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hypercubic

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/lattice/root/hypercubic.yml.