Lattice-shell code[1,2] 


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 [3; 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 — 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 [4,5]. Shells of \(A_n\) and \(D_n\) lattices form infinite families of spherical 3-designs [6; Exam. 2.9].


N. Sloane, “Tables of sphere packings and spherical codes”, IEEE Transactions on Information Theory 27, 327 (1981) DOI
A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. I. Optimum shaping, shell mapping”, IEEE Transactions on Information Theory 39, 1799 (1993) 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.
E. Bannai and E. Bannai, “A survey on spherical designs and algebraic combinatorics on spheres”, European Journal of Combinatorics 30, 1392 (2009) DOI
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Zoo Code ID: lattice_shell

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“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.