Description
A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [3] and sporadic simple groups [4]. Adding a parity bit to the code results in the self-dual \([12,6,6]_3\) extended ternary Golay code. Up to equivalence, both codes are unique for their respective parameters [5]. The dual of the ternary Golay code is a \([11,5,6]_3\) projective two-weight subcode.
A generator matrix for the ternary Golay code is \begin{align} \left(\begin{array}{ccccccccccc} 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}
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 double cover of the Mathieu group \(\mathcal{M}_{12}\), two of the sporadic simple groups.
Decoding
Decoder for the extended ternary Golay code using the tetracode [6].Realizations
Code used in football pools with at least one good bet [7,8]. In fact, the code was originally constructed by Juhani Virtakallio and published in the Finnish football pool magazine Veikkaaja [1,8,9].Proofs of the quantum mechanical Kochen-Specker theorem [10].Cousins
- \([23, 12, 7]\) Golay code
- \([24, 12, 8]\) Extended Golay code
- Self-dual linear code— The extended ternary Golay code is self-dual.
- Projective geometry code— The extended ternary Golay code admits a projective geometric construction ([11], pg. 296).
- Divisible code— Extended ternary Golay code is 3-divisible ([11], 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 [12].
- Projective two-weight code— The dual of the ternary Golay code is a projective two-weight subcode [14,15][13; Exam. 19.3.2][16; Table 7.1].
- \(\Lambda_{24}\) Leech lattice— The Leech lattice can be obtained by from the ternary Golay code [17].
- Combinatorial design— The supports of the weight-five (weight-six) codewords of the (extended) ternary Golay code support the Steiner system \(S(4,5,11)\) (\(S(5,6,12)\)) [18,19][3; pg. 89]. Its blocks are called hexads.
- Pless symmetry code— The Pless symmetry code for \(p=5\) is the extended ternary Golay code.
- Tetracode— Extended ternary Golay codewords can be obtained from tetracodewords [3]. The tetracode can be used to decode the extended ternary Golay code [6].
- \([[11,1,5]]_3\) qutrit Golay code— The qutrit Golay code is a CSS code constructed from the ternary Golay code.
Primary Hierarchy
References
- [1]
- Veikkaus-Lotto (Veikkaaja) magazine, issues 27, 28, and 33, August-September 1947.
- [2]
- M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.
- [3]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
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- [5]
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- [6]
- V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI
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- H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
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- A. E. Brouwer, "Two-weight Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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- N. Tzanakis and J. Wolfskill, “The diophantine equation x2 = 4qa2 + 4q + 1, with an application to coding theory”, Journal of Number Theory 26, 96 (1987) DOI
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- A. E. Brouwer and H. Van Maldeghem, Strongly Regular Graphs (Cambridge University Press, 2022) DOI
- [16]
- R. A. Games, “The packing problem for projective geometries over GF(3) with dimension greater than five”, Journal of Combinatorial Theory, Series A 35, 126 (1983) DOI
- [17]
- “Twenty-three constructions for the Leech lattice”, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 381, 275 (1982) DOI
- [18]
- L. J. Paige, “A Note on the Mathieu Groups”, Canadian Journal of Mathematics 9, 15 (1957) DOI
- [19]
- M. HUBER, “CODING THEORY AND ALGEBRAIC COMBINATORICS”, Selected Topics in Information and Coding Theory 121 (2010) arXiv:0811.1254 DOI
- [20]
- P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) 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