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

Cousins

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
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Zoo Code ID: best

Cite as:
“Best \((10,40,4)\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/best
BibTeX:
@incollection{eczoo_best,
  title={Best \((10,40,4)\) code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/best}
}
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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.