Best \((10,40,4)\) code

Best \((10,40,4)\) code[1,2]

Description

Binary nonlinear \((10,40,4)\) code that is unique [3]. Under Construction \(A\), this code yields \(P_{10c}\), a non-lattice sphere packing that is the densest known in 10 dimensions [4][5; pg. 140]. Codewords can be obtained by applying the Gray map to a nonlinear code over \(\mathbb{Z}_4\) called the pentacode [6; Sec. 2].

Parent

Binary code

Cousins

Construction-\(A\) code — Using Construction \(A\), the Best code yields \(P_{10c}\), a non-lattice sphere packing in 10 dimensions that is the densest known [4][5; pg. 140].
Sphere packing — Using Construction \(A\), the Best code yields \(P_{10c}\), a non-lattice sphere packing in 10 dimensions that is the densest known [4][5; pg. 140].
\(q\)-ary code over \(\mathbb{Z}_q\) — The Best code is an image of a nonlinear code over \(\mathbb{Z}_4\) under the Gray map [6; Sec. 2].

References

[1]: Best, M.R. 1978. Binary codes with minimum distance four. Report ZW 112/78, Math Centrum, Amsterdam.
[2]: M. Best, “Binary codes with a minimum distance of four (Corresp.)”, IEEE Transactions on Information Theory 26, 738 (1980) DOI
[3]: S. Litsyn and A. Vardy, “The uniqueness of the Best code”, IEEE Transactions on Information Theory 40, 1693 (1994) DOI
[4]: J. Leech and N. J. A. Sloane, “Sphere Packings and Error-Correcting Codes”, Canadian Journal of Mathematics 23, 718 (1971) DOI
[5]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[6]: J. H. Conway and N. J. A. Sloane, “Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others”, Designs, Codes and Cryptography 4, 31 (1994) DOI

Page edit log

Victor V. Albert (2022-11-23) — most recent

Cite as:

“Best \((10,40,4)\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/best

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/nonlinear/sphere_packing/best.yml.