## 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 [3; Ch. 10] for tables of lattice-shell codes.

## Parent

## Children

- \(BW_{32}\) lattice-shell code
- \(D_4\) lattice-shell code
- \(E_8\) Gosset lattice-shell code
- \(E_7\) lattice-shell code
- \(E_6\) lattice-shell code
- \(\Lambda_{16}\) lattice-shell code
- \(\Lambda_{24}\) Leech lattice-shell code
- Cubeoctahedron code — Cubeoctahedron codewords form the minimal shell of the \(D_3\) face-centered cubic (fcc) lattice.
- Biorthogonal spherical code — Biorthogonal 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.

## 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 — 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].

## References

- [1]
- N. Sloane, “Tables of sphere packings and spherical codes”, IEEE Transactions on Information Theory 27, 327 (1981) DOI
- [2]
- A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. I. Optimum shaping, shell mapping”, IEEE Transactions on Information Theory 39, 1799 (1993) DOI
- [3]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [4]
- B. B. Venkov. Even unimodular extremal lattices, algebraic geometry and its applications. Trudy Mat. Inst. Steklov., 165:43–48, 1984.
- [5]
- 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.
- [6]
- E. Bannai and E. Bannai, “A survey on spherical designs and algebraic combinatorics on spheres”, European Journal of Combinatorics 30, 1392 (2009) DOI

## Page edit log

- Victor V. Albert (2022-11-16) — most recent

## Cite as:

“Lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/lattice_shell