Lattice-shell code[1] 


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.


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




  • 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.


N. Sloane, “Tables of sphere packings and spherical codes”, IEEE Transactions on Information Theory 27, 327 (1981) DOI
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
B. B. Venkov. Even unimodular extremal lattices, algebraic geometry and its applications. Trudy Mat. Inst. Steklov., 165:43–48, 1984.
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.
@incollection{eczoo_lattice_shell, title={Lattice-shell code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.