\(E_8\) Gosset lattice code
BW lattice in dimension \(8\), which is the lattice corresponding to the \([8,4,4]\) Hamming code via the mod-two lattice construction.
The \(E_8\) lattice has a nominal coding gain of \(2\). It exhibits the densest lattice packing [2–5], densest packing , and highest kissing number of 240  in eight dimensions.
Popular summary of solution to the sphere-packing problem in Quanta Magazine.
- \([8,4,4]\) extended Hamming code — The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice code via the mod-two lattice construction [9; Ex. 10.5.2].
- Octacode — The octacode yields the \(E_8\) Gosset lattice code via Construction \(A_4\) [10,11].
- \(q\)-ary Hamming code — The \([4,2,3]_3\) ternary Hamming code can be used to obtain the \(E_8\) Gosset lattice code [9; Ex. 10.5.5].
- Repetition code — The \([8,1,8]\) repetition code can be used to obtain the \(E_8\) Gosset lattice code [9; Ex. 10.7.1].
- Spherical sharp configuration — Several spherical sharp configrations are derived from the \(E_8\) Gosset lattice code .
- \(E_8\) Gosset lattice-shell code
- Gosset, Thorold. "On the regular and semi-regular figures in space of n dimensions." Messenger of Mathematics 29 (1900): 43-48.
- Blichfeldt, H. F. "On the minimum value of positive real quadratic forms in 6 variables." Bulletin of American Math. Soc 31 (1925): 386.
- H. F. Blichfeldt, “The minimum value of quadratic forms, and the closest packing of spheres”, Mathematische Annalen 101, 605 (1929) DOI
- H. F. Blichfeldt, “The minimum values of positive quadratic forms in six, seven and eight variables”, Mathematische Zeitschrift 39, 1 (1935) DOI
- G. L. Watson, “The Class-Number of a Positive Quadratic Form”, Proceedings of the London Mathematical Society s3-13, 549 (1963) DOI
- H. Cohn et al., “The sphere packing problem in dimension \(24\)”, Annals of Mathematics 185, (2017) arXiv:1603.06518 DOI
- Vetchinkin, N. M. "Uniqueness of classes of positive quadratic and highest kissing number of 240 in eight dimensions"
- H. Cohn et al., “Universal optimality of the \(E_8\) and Leech lattices and interpolation formulas”, (2022) arXiv:1902.05438
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Algebraic Coding 194 (1994) DOI
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
Page edit log
- Victor V. Albert (2022-11-08) — most recent
“\(E_8\) Gosset lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/eeight