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 [3–5].
- \(\Lambda_{24}\) Leech lattice— The Leech lattice can be constructed from the quaternary Golay code [1,2].
Member of code lists
Primary Hierarchy
Parents
The quaternary Golay code is an extremal Type II self-dual code over \(\mathbb{Z}_4\) by virtue of its parameters [6].
Quaternary Golay code
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
- [6]
- A. Munemasa and R. A. L. Betty, “Classification of extremal type II \(\)\mathbb {Z}_4\(\)-codes of length 24”, Designs, Codes and Cryptography 92, 771 (2023) DOI
Page edit log
- Victor V. Albert (2025-03-26) — most recent
Cite as:
“Quaternary Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/quaternary_golay