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Construction A code[1]

Alternative names: Mod-two lattice.

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.

Cousins

Member of code lists

Primary Hierarchy

Parents
Construction A code
Children
\([n,n-1,2]\) SPC codes yield \(D_n\) checkerboard lattices via Construction A [4; Exam. 10.5.1].
The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice via Construction A [4; Exam. 10.5.2].
The \([7,3,4]\) simplex code yields the \(E_7\) root lattice via Construction A [5][4; Exam. 10.5.3].

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]
J. Leech and N. J. A. Sloane, “Sphere Packings and Error-Correcting Codes”, Canadian Journal of Mathematics 23, 718 (1971) DOI
[4]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[5]
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
Page edit log

Cite as:

“Construction A code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/construction_a

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/sphere_packing/from_codes/construction_a.yml.