Extended quaternary Golay code

Extended quaternary Golay code[1]

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

Description

An extended quadratic residue quaternary linear \((24,4^{12},12)_{\mathbb{Z}_4}\) code 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 extended 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]. The quaternary Golay code can be constructed from the extended Golay code by Hensel lifting to \(\mathbb{Z}_4\) [4][3; 3rd Ed., pg. xxxiii].
Combinatorial design— Supports of codewords of any fixed symmetrized type of the extended quaternary Golay code form a 5-design [5–7].
\(\Lambda_{24}\) Leech lattice— The Leech lattice can be constructed from the extended quaternary Golay code via Construction \(A_4\) [3; 3rd Ed., pg. xxxiii] (see also [1,2,8]).

Member of code lists

Primary Hierarchy

Classical Domain

Ring Kingdom

Ring codeECC

\(q\)-ary code over \(\mathbb{Z}_q\)ECC

Linear code over \(\mathbb{Z}_q\)ECC

Linear code over \(\mathbb{Z}_4\)

Dual code over \(\mathbb{Z}_4\)Linear code over \(\mathbb{Z}_q\) ECC

Self-dual code over \(\mathbb{Z}_4\)Linear code over \(\mathbb{Z}_q\) ECC

Parents

Self-dual code over \(\mathbb{Z}_4\)

The extended quaternary Golay code is an extremal Type II self-dual code over \(\mathbb{Z}_4\) by virtue of its parameters [9].

Extended 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”, Communications in Mathematical Physics 406, (2025) arXiv:2309.02382 DOI
[3]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[4]: A. R. Calderbank and N. J. A. Sloane, “Modular and p-adic cyclic codes”, (2003) arXiv:math/0311319
[5]: 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
[6]: M. Harada, “New 5-designs constructed from the lifted Golay code over ?4”, Journal of Combinatorial Designs 6, 225 (1998) DOI
[7]: A. Bonnecaze, E. Rains, and P. Solé, “3-Colored 5-Designs and Z4-Codes”, Journal of Statistical Planning and Inference 86, 349 (2000) DOI
[8]: “Twenty-three constructions for the Leech lattice”, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 381, 275 (1982) DOI
[9]: 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:

“Extended 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/self_dual/quaternary_golay.yml.