Description
Lattice-based \(n\)-dimensional code whose codewords form the dual of the \(A_n\) lattice.Protection
Exhibits the thinnest covering in two dimensions and the thinnest lattice covering in dimensions three [1], four [2], and five [3,4].Cousin
Member of code lists
Primary Hierarchy
Parents
\(A_n^{\perp}\) lattice
Children
The bcc lattice is the dual of the \(A_3=D_3\) fcc lattice.
The \(A_2\) lattice is equivalent, up to rescaling, to its dual \(A_2^{\perp}\) [5; Ch. 1, pg. 13].
References
- [1]
- R. P. Bambah and H. Gupta, “On lattice coverings by spheres.” Proceedings of the National Institute of Sciences of India. Vol. 20. Indian National Science Academy, 1954.
- [2]
- B. N. Delaunay and S. S. Ryskov, “Solution of the problem of least dense lattice covering of a four-dimensional space by equal spheres.” Soviet Mathematics Doklady Vol. 4. 1963.
- [3]
- S. S. Ryshkov and E. P. Baranovskii, “Solution of the problem of least dense lattice covering of five-dimensional space by equal spheres”, Doklady Akademii Nauk SSSR, 222:1 (1975), 39–42.
- [4]
- S. S. Ryshkov and E. P. Baranovskii, “C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)”, Trudy Matematicheskogo Instituta imeni V. A. Steklova, 137, 1976, 3–131; Proceedings of the Steklov Institute of Mathematics, 137 (1976), 1–140.
- [5]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
Page edit log
- Victor V. Albert (2022-02-26) — most recent
Cite as:
“\(A_n^{\perp}\) lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/an_dual