Construction-\(A\) code[1] 

Description

Sphere packing constructed from a binary \((n,K)\) code using Construction \(A\) [2]. 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

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 \(A\).
  • 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 [3][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]
J. Leech and N. J. A. Sloane, “Sphere Packings and Error-Correcting Codes”, Canadian Journal of Mathematics 23, 718 (1971) DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: construction_a

Cite as:
“Construction-\(A\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/construction_a
BibTeX:
@incollection{eczoo_construction_a, title={Construction-\(A\) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/construction_a} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/construction_a

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/lattice/construction_a.yml.