Lattice-shell code[1] 

Description

Spherical code whose codewords are scaled versions of points on a lattice. A \(m\)-shell code consists of normalized lattice vectors \(x\) with squared norm \(\|x\|^2 = m\). Each code is constructed by normalizing a set of lattice vectors in one or more shells, i.e., sets of lattice points lying on a hypersphere.

Notes

See [2; Ch. 10] for tables of lattice-shell codes.

Parent

Children

Cousins

  • Lattice-based code — Lattice-shell codes consists of lattice points that have been normalized.
  • Cyclic code — Lattice-shell codewords are often permutations of a particular set of reference vectors, meaning that a cyclic permutation of a codeword yields another codeword.
  • Spherical design code — Nonempty \(2m\)-shell codes of extremal even unimodular lattices in \(n\) dimensions form spherical \(t\)-designs with \(t=11\) (\(t=7\), \(t=3\)) if \(n \equiv 0\) (\(n \equiv 8\), \(n\equiv 16\)) modulo 24 [3,4].
  • Hessian polyhedron code — Double Hessian polyhedron codewords form the minimal lattice-shell code of the \(E_6^{\perp}\) lattice.

References

[1]
N. Sloane, “Tables of sphere packings and spherical codes”, IEEE Transactions on Information Theory 27, 327 (1981) DOI
[2]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[3]
B. B. Venkov. Even unimodular extremal lattices, algebraic geometry and its applications. Trudy Mat. Inst. Steklov., 165:43–48, 1984.
[4]
B. Venkov. Réseaux et designs sphériques. In Réseaux euclidiens, designs sphériques et formes modulaires, volume 37 of Monogr. Enseign. Math., pages 10–86. Enseignement Math., Geneva, 2001.
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Zoo Code ID: lattice_shell

Cite as:
“Lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/lattice_shell
BibTeX:
@incollection{eczoo_lattice_shell, title={Lattice-shell code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/lattice_shell} }
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“Lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/lattice_shell

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/lattice_shell/lattice_shell.yml.