Description
Self-dual ternary code that corrects two errors on twelve trits and is unique for its parameters up to equivalence [1]. It is obtained from the \([11,6,5]_3\) ternary Golay code by adding a parity-check trit.
Its weight enumerator is the Gleason polynomial \(g_5\) [2; Rem. 4.2.6]. Its monomial automorphism group is the double cover \(2.\mathcal{M}_{12}\) of the Mathieu group \(\mathcal{M}_{12}\), a sporadic simple group.
Decoding
Decoder using the tetracode [3].Cousins
- \([11,6,5]_3\) Ternary Golay code— Adding a parity-check trit to the ternary Golay code yields the extended ternary Golay code.
- \([24, 12, 8]\) Extended Golay code— The extended ternary Golay code is the ternary counterpart of the extended Golay code.
- Combinatorial design— The supports of the weight-six codewords of the extended ternary Golay code form the Steiner system \(S(5,6,12)\), whose blocks are called hexads [4,5][6; pg. 89].
- Niemeier lattice— The extended ternary Golay code is the glue code for the Niemeier lattice \(A^{12}_2\) [6; Ch. 16, pg. 408].
- \([4,2,3]_3\) Tetracode— Extended ternary Golay codewords can be obtained from tetracodewords [6]. The tetracode can be used to decode the extended ternary Golay code [3].
Primary Hierarchy
\([2q+2,q+1]_3\) Pless symmetry codeDivisible Linear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC
Parents
The Pless symmetry code for \(q=5\) is equivalent to the extended ternary Golay code [7; Ch. 16, Prob. 18].
The extended ternary Golay code is a ternary sharp configuration [8; Table 12.1].
The extended ternary Golay code is projective because its dual, the code itself, has distance six [7; pg. 178].
\([12,6,6]_3\) Extended ternary Golay code
References
- [1]
- P. Delsarte and J. M. Goethals, “Unrestricted codes with the golay parameters are unique”, Discrete Mathematics 12, 211 (1975) DOI
- [2]
- S. Bouyuklieva, “Self-dual codes”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 79-96 DOI
- [3]
- V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI
- [4]
- L. J. Paige, “A Note on the Mathieu Groups”, Canadian Journal of Mathematics 9, 15 (1957) DOI
- [5]
- M. HUBER, “CODING THEORY AND ALGEBRAIC COMBINATORICS”, Series on Coding Theory and Cryptology 121 (2010) arXiv:0811.1254 DOI
- [6]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [7]
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
- [8]
- P. Boyvalenkov, D. Danev, “Linear programming bounds”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 251-266 DOI
Page edit log
- Victor V. Albert (2026-10-13) — most recent
Cite as:
“\([12,6,6]_3\) Extended ternary Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/extended_ternary_golay, arXiv:2606.11484