Tetracode[1]
Description
The \([4,2,3]_3\) ternary self-dual MDS code that has connections to lattices [1].
A generator matrix is \begin{align} \begin{pmatrix}1 & 0 & 1 & 1\\ 0 & 1 & 1 & 2 \end{pmatrix}~, \tag*{(1)}\end{align} where \(GF(3) = \{0,1,2\}\).
Notes
See corresponding MinT database entry [2].Cousin
- 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 [3].
Member of code lists
- \(q\)-ary linear codes
- Algebraic-geometry codes
- Classical codes
- Constant-weight codes
- Evaluation codes
- Locally correctable codes
- Locally decodable codes
- Locally recoverable codes
- MDS codes
- Orthogonal arrays and friends
- Perfect codes
- Projective codes
- Self-dual classical codes and friends
- Small-distance classical codes
- Universally optimal codes
Primary Hierarchy
\(q\)-ary simplex codeMDS Linear \(q\)-ary OA LRC Distributed-storage \(t\)-design Universally optimal Constant-weight ECC
Parents
The tetracode is equivalent to \(S(3,2)\).
The tetracode is equivalent to the \(r=2\) \(3\)-ary Hamming code.
The tetracode is Euclidean self-dual.
Maximum distance separable (MDS) codeLinear \(q\)-ary OA LRC Distributed-storage \(t\)-design Universally optimal ECC
Extended GRS codeGRM GRS Evaluation MDS Linear \(q\)-ary OA AG \(t\)-design Universally optimal LDC LCC LRC Distributed-storage ECC
The tetracode is an extended RS code [1; pg. 81].
The tetracode is a lexicode [4].
Tetracode
References
- [1]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [2]
- Rudolf Schürer and Wolfgang Ch. Schmid. “Tetracode.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CTetra.html
- [3]
- V. Pless, “Decoding the Golay codes”, IEEE Transactions on Information Theory 32, 561 (1986) DOI
- [4]
- J. Conway and N. Sloane, “Lexicographic codes: Error-correcting codes from game theory”, IEEE Transactions on Information Theory 32, 337 (1986) DOI
Page edit log
- Victor V. Albert (2022-08-09) — most recent
Cite as:
“Tetracode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/tetracode