Ternary Golay code[1] 

Description

A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [2] and sporadic simple groups [3]. Adding a parity bit to the code results in the \([12, 6, 6]\) extended ternary Golay code. Up to equivalence, both codes are unique for their respective parameters.

A generator matrix for the ternary Golay code is \begin{align} \left(\begin{array}{cccccccccccc} 1 & 0 & 0 & 0 & 0 & | & 1 & 1 & 1 & 2 & 2 & 0 \\ 0 & 1 & 0 & 0 & 0 & | & 1 & 1 & 2 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 & 0 & | & 1 & 2 & 1 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 & 0 & | & 1 & 2 & 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 & 1 & | & 1 & 0 & 2 & 2 & 1 & 1 \\ \end{array}\right)~. \tag*{(1)}\end{align}

Decoding

Decoder for the extended ternary Golay code using the tetracode [4].

Realizations

Code used in football pools with at least one good bet [5,6]. In fact, the code was originally constructed by Juhani Virtakallio and published in the Finnish football pool magazine Veikkaaja [6,7].Proofs of the quantum mechanical Kochen-Specker theorem [8].

Notes

The automorphism group of the ternary Golay code is the Mathieu group \(\mathcal{M}_{11}\), and the automorphism group of the extended ternary Golay code is the Mathieu group \(\mathcal{M}_{12}\), two of the sporadic simple groups.

Parents

Cousins

References

[1]
M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.
[2]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[3]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[4]
V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI
[5]
H. Hämäläinen and S. Rankinen, “Upper bounds for football pool problems and mixed covering codes”, Journal of Combinatorial Theory, Series A 56, 84 (1991) DOI
[6]
A. Barg, “At the Dawn of the Theory of Codes”, The Mathematical Intelligencer 15, 20 (1993) DOI
[7]
T. Thompson, From Error-Correcting Codes Through Sphere Packings to Simple Groups (American Mathematical Society, 1983) DOI
[8]
M. Waegell and P. K. Aravind, “Golay codes and quantum contextuality”, Physical Review A 106, (2022) arXiv:2206.04209 DOI
[9]
P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[10]
J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
[11]
H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
[12]
A. E. Brouwer, "Two-weight Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[13]
10.1007/978-1-4757-6568-7
Page edit log

Your contribution is welcome!

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

edit on this site

Zoo Code ID: ternary_golay

Cite as:
“Ternary Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ternary_golay
BibTeX:
@incollection{eczoo_ternary_golay, title={Ternary Golay code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/ternary_golay} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/ternary_golay

Cite as:

“Ternary Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ternary_golay

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/easy/ternary_golay.yml.