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Quaternary Golay code[1]

Alternative names: \(\mathbb{Z}_4\) Golay code.

Description

An extended quadratic residue quaternary linear code over \(\mathbb{Z}_4\) that is a quaternary version of the Golay code. The code has Lee distance 12, Hamming distance 8, and Euclidean distance 16 [1]. The code maps to a binary code under the Gray map whose distance distribution is equal to the MacWilliams transform of the distance distribution of its dual code.

Cousins

  • \([24, 12, 8]\) Extended Golay code— Codewords of the quaternary Golay code with entries 0 and 2 are of the form \(2c\), where \(c\) is a codeword of the extended Golay code. Its mod-two reduction (mapping \(0,1,2,3\) to \(0,1,0,1\)) also yields the extended Golay code; see Ref. [2].
  • Combinatorial design— Supports of codewords of any fixed symmetrized type of the quaternary Golay code form a 5-design [35].
  • \(\Lambda_{24}\) Leech lattice— The Leech lattice can be constructed from the quaternary Golay code [1,2].

References

[1]
A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
[2]
G. W. Moore and R. K. Singh, “Beauty And The Beast Part 2: Apprehending The Missing Supercurrent”, (2023) arXiv:2309.02382
[3]
T. A. Gulliver and M. Harada, “Extremal double circulant Type II codes over Z4 and construction of 5-(24, 10, 36) designs”, Discrete Mathematics 194, 129 (1999) DOI
[4]
M. Harada, “New 5-designs constructed from the lifted Golay code over ?4”, Journal of Combinatorial Designs 6, 225 (1998) DOI
[5]
A. Bonnecaze, E. Rains, and P. Solé, “3-Colored 5-Designs and Z4-Codes”, Journal of Statistical Planning and Inference 86, 349 (2000) DOI
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Zoo Code ID: quaternary_golay

Cite as:
“Quaternary Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/quaternary_golay
BibTeX:
@incollection{eczoo_quaternary_golay, title={Quaternary Golay code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quaternary_golay} }
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Permanent link:
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Cite as:

“Quaternary Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/quaternary_golay

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/over_z4/linear_over_z4/quaternary_golay.yml.