Coxeter-Todd \(K_{12}\) lattice[1]
Description
Even integral lattice in dimension \(12\) that gives the densest known lattice packing. Its automorphism group was discovered by Mitchell [2]. As a real lattice, \(K_{12}\) is equivalent, up to rescaling, to its dual \(K_{12}^{\perp}\) [3; Ch. 4, pg. 128]. For more details, see [4][3; Sec. 4.9].Protection
The \(K_{12}\) lattice exhibits the densest known lattice packing in 12 dimensions.Cousins
- \(\Lambda_{24}\) Leech lattice— The Coxeter-Todd lattice can be realized as a subset of the Leech lattice [3; Ch. 4, pg. 128].
- \([6,3,4]_4\) Hexacode— The hexacode can be used to obtain the Coxeter-Todd \(K_{12}\) lattice [5; Exam. 10.5.6][3; Ch. 7, pg. 198].
- \(q\)-ary repetition code— Applying Construction \(B_c\) to the ternary repetition code of length \(6\) over the Eisenstein integers yields the Coxeter-Todd \(K_{12}\) lattice [3; Ch. 7, pg. 200].
- Unimodular lattice— The Coxeter-Todd lattice is unimodular when defined as a complex lattice over the Eisenstein integers [3; Ch. 4, pg. 128].
Member of code lists
Primary Hierarchy
Parents
Coxeter-Todd \(K_{12}\) lattice
References
- [1]
- H. S. M. Coxeter and J. A. Todd, “An Extreme Duodenary Form”, Canadian Journal of Mathematics 5, 384 (1953) DOI
- [2]
- H. H. Mitchell, “Determination of All Primitive Collineation Groups in More than Four Variables which Contain Homologies”, American Journal of Mathematics 36, 1 (1914) DOI
- [3]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [4]
- J. H. Conway and N. J. A. Sloane, “The Coxeter–Todd lattice, the Mitchell group, and related sphere packings”, Mathematical Proceedings of the Cambridge Philosophical Society 93, 421 (1983) DOI
- [5]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
Page edit log
- Victor V. Albert (2022-12-12) — most recent
Cite as:
“Coxeter-Todd \(K_{12}\) lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/coxeter_todd