\([24, 12, 8]\) Extended Golay code[1]
Description
A self-dual \([24, 12, 8]\) code that is obtained from the Golay code by adding a parity check. Up to equivalence, it is unique for its parameters [2].
The automorphism group of the extended Golay code is the Mathieu group \(\mathcal{M}_{24}\), a sporadic simple group.
Decoding
Majority decoding [3].Decoder using the hexacode [4].The extended Golay code has a trellis representation and can thus be decoded using trellis decoding [5,6].Realizations
Voyager 1 and 2 spacecraft, transmitting hundreds of color pictures of Jupiter and Saturn in their 1979, 1980, and 1981 fly-bys [7].American military standards for automatic link establishment in high frequency radio systems [8].Cousins
- Icosahedron code— The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see post by J. Baez for more details. To construct the code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.
- Dodecahedron code— The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see post by J. Baez for more details. To construct the code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.
- \([23, 12, 7]\) Golay code— The extended Golay code is an extension of the Golay code by a parity-check bit.
- Combinatorial design— The supports of the weight-eight codewords of the extended Golay code support the Steiner system \(S(5,6,12)\) [9,10][11; pg. 89]. Its blocks are called octads.
- Nordstrom-Robinson (NR) code— The NR code can be constructed using the extended Golay code by first selecting a set of codewords satisfying certain conditions and then deleting specific coordinates [12; pg. 73].
- Hexacode— Extended Golay codewords can be obtained from hexacodewords [11]. The hexacode can be used to decode the extended Golay code [4].
- Ternary Golay code
Primary Hierarchy
Parents
The extended Golay code is nearly perfect.
The extended Golay code is equivalent to the Karlin double circulant code for \(m=11\) [12; Ch. 16].
The extended Golay code is the unique code at its parameters and happens to be self-dual and doubly even [2][13; Remark 4.3.11].
The extended Golay code is an extended binary quadratic-residue code [12; Ch. 16].
The extended Golay code is an orthogonal array of strength 7 [14; Exam. 1].
The extended Golay code is a group-algebra code for various groups [15–17]; see [18; Exam. 16.5.1].
The extended Golay code is a lexicode [19,20][12; pg. 327].
The Golay code and several of its extended, shortened, and punctured versions are LP universally optimal codes [21].
\([24, 12, 8]\) Extended Golay code
References
- [1]
- M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.
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- P. Delsarte and J. M. Goethals, “Unrestricted codes with the golay parameters are unique”, Discrete Mathematics 12, 211 (1975) DOI
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- A. J. VITERBI, “Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm”, The Foundations of the Digital Wireless World 41 (2009) DOI
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- B. Honary and G. Markarian, “New simple encoder and trellis decoder for Golay codes”, Electronics Letters 29, 2170 (1993) DOI
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- E. C. Stone, “The Voyager 2 encounter with Uranus”, Journal of Geophysical Research: Space Physics 92, 14873 (1987) DOI
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- E. E. Johnson. An Efficient Golay Codec For MIL-STD-188-141A and FED-STD-1045. Department of Electrical and Computer Engineering, New Mexico State University, 1991.
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- E. M. Rains and N. J. A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp. 177–294.
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- P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory”, IEEE Transactions on Information Theory 44, 2477 (1998) DOI
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- I. McLoughlin and T. Hurley, “A Group Ring Construction of the Extended Binary Golay Code”, IEEE Transactions on Information Theory 54, 4381 (2008) DOI
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- S. T. Dougherty, J. Gildea, R. Taylor, and A. Tylyshchak, “Group rings, G-codes and constructions of self-dual and formally self-dual codes”, Designs, Codes and Cryptography 86, 2115 (2017) DOI
- [17]
- F. Bernhardt, P. Landrock, and O. Manz, “The extended golay codes considered as ideals”, Journal of Combinatorial Theory, Series A 55, 235 (1990) DOI
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- W. Willems, "Codes in Group Algebras." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [19]
- M. J. T. Guy, unpublished.
- [20]
- J. Conway and N. Sloane, “Lexicographic codes: Error-correcting codes from game theory”, IEEE Transactions on Information Theory 32, 337 (1986) DOI
- [21]
- 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 (2025-01-07) — most recent
Cite as:
“\([24, 12, 8]\) Extended Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/extended_golay