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\([4,2,3]_3\) Tetracode [1]

Description

The \([4,2,3]_3\) ternary self-dual MDS code that has connections to lattices [1]. It is the unique MDS code with its parameters [2,3]. Its weight enumerator is the Gleason polynomial \(g_4\) [4; Rem. 4.2.6].

A generator matrix is \begin{align} \begin{pmatrix}1 & 0 & 1 & 1\\ 0 & 1 & 1 & 2 \end{pmatrix}~,\tag*{$(1)$} \end{align}where \(\mathbb{F}_3 = \{0,1,2\}\).

Notes

See corresponding MinT database entry [5].

Cousins

  • \([12,6,6]_3\) Extended ternary Golay code— Extended ternary Golay codewords can be obtained from tetracodewords [1]. The tetracode can be used to decode the extended ternary Golay code [6].
  • \(E_8\) Gosset lattice— The \([4,2,3]_3\) tetracode can be used to obtain the \(E_8\) Gosset lattice [7; Exam. 10.5.5][1; Ch. 7, pg. 200].
  • Niemeier lattice— The tetracode is the glue code for the Niemeier lattice \(E_6^4\) [1; Ch. 16, pg. 408].

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Dual linear codeECC
Self-dual linear codeSelf-dual additive ECC
Parents
Self-dual linear code
The tetracode is Euclidean self-dual, i.e., Type III in the terminology of [4; Rems. 4.2.6 and 4.3.2].
Self-dual code over \(\mathbb{Z}_q\)Linear code over \(\mathbb{Z}_q\) ECC
\(q\)-ary simplex codeGRM Evaluation Projective geometry Linear \(q\)-ary LCC LRC Distributed-storage AG Universally optimal ECC
The tetracode is equivalent to \(S(3,2)\).
\(q\)-ary Hamming codeLinear \(q\)-ary Perfect Universally optimal Small-distance block ECC
The tetracode is equivalent to the \(r=2\) \(3\)-ary Hamming code.
Extended GRS codeGRM Evaluation MDS Linear \(q\)-ary LCC LRC Distributed-storage AG OA Universally optimal ECC \(t\)-design
The tetracode is an extended RS code [1; pg. 81].
Lexicographic codeECC
The tetracode is a lexicode [8].
\([4,2,3]_3\) Tetracode

References

  1. [1]

    J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI

  2. [2]

    O. Taussky and J. Todd, “Covering theorems for groups”, Bulletin of the American Mathematical Society 54(3) (1948)

  3. [3]

    J. G. Kalbfleisch and R. G. Stanton, “A Combinatorial Problem in Matching”, Journal of the London Mathematical Society s1-44, 60 (1969) DOI

  4. [4]

    S. Bouyuklieva, “Self-dual codes”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 79-96 DOI

  5. [5]

    R. Schürer and W. Ch. Schmid, “Tetracode”, From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03 URL

  6. [6]

    V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI

  7. [7]

    T. Ericson and V. A. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)

  8. [8]

    J. Conway and N. Sloane, “Lexicographic codes: Error-correcting codes from game theory”, IEEE Transactions on Information Theory 32, 337 (1986) DOI

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Zoo Code ID: tetracode

Cite as:
“\([4,2,3]_3\) Tetracode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/tetracode, arXiv:2606.11484
BibTeX:
@incollection{eczoo_tetracode,
title={\([4,2,3]_3\) Tetracode},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/tetracode}
}
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Cite as:

“\([4,2,3]_3\) Tetracode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/tetracode, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/easy/tetracode.yml.