Construction-\(A\) code[1]
Also known as Mod-two lattice code.
Description
Sphere packing constructed from a binary \((n,K)\) code using Construction A [2].
Construction A: Construction \(A\) converts a linear binary code into a sphere packing. Each binary codeword \(c\) of the code is mapped to an infinite set of points \(x\) such that \(x = c\) modulo two. If the underlying binary code is linear, then the resulting set of points forms a lattice.
Parent
Children
- \(D_n\) checkerboard lattice code — \([n,n-1,2]\) SPC codes yield \(D_n\) checkerboard lattice codes via Construction A [3; Ex. 10.5.1].
- \(E_8\) Gosset lattice code — The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice code via Construction A [3; Ex. 10.5.2].
- \(E_7\) root lattice code — The \([7,4,3]\) Hamming code yields the \(E_7^{\perp}\) root lattice code via Construction A [4]. The \([7,3,4]\) little Hamming code yields the \(E_7\) root lattice code via the same construction [4][3; Ex. 10.5.3].
Cousins
- Binary code — Each binary code yields a sphere packing under Construction A.
- Linear binary code — Every binary linear code yields a lattice code under Construction A.
- \(D_4\) hyper-diamond lattice code — The \(D_4\) lattice is constructed out of the \([4,1,4]\) repetition and \([4,3,2]\) SPC codes via the Construction A.
- Best \((10,40,4)\) code — Using Construction A, the Best code yields \(P_{10c}\), a non-lattice sphere packing in 10 dimensions that is the densest known [5][2; pg. 140].
- Julin-Golay code — Using Construction A, the Julin-Golay codes yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.
References
- [1]
- J. Leech, “Notes on Sphere Packings”, Canadian Journal of Mathematics 19, 251 (1967) DOI
- [2]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [3]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [4]
- J. H. Conway and N. J. A. Sloane, “On the Voronoi Regions of Certain Lattices”, SIAM Journal on Algebraic Discrete Methods 5, 294 (1984) DOI
- [5]
- J. Leech and N. J. A. Sloane, “Sphere Packings and Error-Correcting Codes”, Canadian Journal of Mathematics 23, 718 (1971) DOI
Page edit log
- Victor V. Albert (2022-11-08) — most recent
Cite as:
“Construction-\(A\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/construction_a