Ternary Golay code

Ternary Golay code[1]

Description

A \([11,6,5]_{GF(3)}\) 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

\(q\)-ary quadratic-residue (QR) code — The ternary Golay code is a quadratic residue code over \(GF(3)\) with residue set \(Q = \{1, 3, 4, 5, 9\} \) and generator polynomial \(x^5 + x^4 - x^3 + x^2 - 1\) ([3], Ch. 16).
Perfect code — The ternary Golay code is perfect.
\(q\)-ary sharp configuration — The ternary Golay code and one of its shortened versions are \(q\)-ary sharp configurations [9; Table 12.1].

Cousins

Golay code
Dual linear code — The extended ternary Golay code is self-dual.
Projective geometry code — The extended ternary Golay code admits a projective geometric construction ([10], pg. 296).
Divisible code — Extended ternary Golay code is 3-divisible ([10], pg. 296).
Universally optimal \(q\)-ary code — The ternary Golay code and several of its extended, shortened, and punctured versions are LP universally optimal codes [11].
Tetracode — Extended ternary Golay codewords can be obtained from tetracodewords [2]. The tetracode can be used to decode the extended ternary Golay code [4].
Pless symmetry code — The Pless symmetry code for \(p=5\) is the extended ternary Golay code.

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. M. Thompson, From Error-Correcting Codes Through Sphere Packings To Simple Groups, Mathematical Association of America, 1983.
[8]: M. Waegell and P. K. Aravind, “Golay codes and quantum contextuality”, Physical Review A 106, (2022) arXiv:2206.04209 DOI
[9]: W. C. Huffman, J.-L. Kim, and P. Solé, 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

Page edit log

Vikram Elijah Amin (2023-01-21) — most recent
Victor V. Albert (2022-01-21)
Victor V. Albert (2022-08-12)
Shashank Sule (2022-03-02)

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/tree/main/codes/classical/q-ary_digits/easy/ternary_golay.yml.