Dodecacode[1]
Description
The unique trace-Hermitian self-dual additive \((12,4^6,6)_4\) code. Its codewords are cyclic permutations of \((\omega 10100100101)\), where \(GF(4)=\{0,1,\omega,\bar{\omega}\}\) is the quaternary Galois field [2; Sec. 2.4.8]. Another generator matrix can be found in [3; Exam. 9.10.8].
The dodecacode is a self-dual additive code, and there is no self-dual linear code with the same parameters [4].
Puncturing the code yields the \((11,4^6,5)_4\) additive code known as the punctured or shortened dodecacode [5].
Cousins
- Combinatorial design— There exists a \(5\)-\((12, 6, 3)\) design in the dodecacode, and a \(3\)-\((11, 5, 4)\) design in the shortened dodecacode [6].
- \([[11,1,5]]\) quantum dodecacode— The dodecacode corresponds to a \([[12,0,6]]\) quantum code in the \(GF(4)\) representation [1]. The \([[11,1,5]]\) quantum dodecacode code corresponds to the shortened dodecacode [7]. A pure \([[10,1,4]]\) quantum code can be obtained from the doubly punctured dodecacode [7]. These codes are not obtained from the Hermitian construction since none of the classical codes are linear.
- Uniformly packed code— The punctured dodecacode code is uniformly packed [8].
Member of code lists
Primary Hierarchy
Parents
The dodecacode is trace-Hermitian self-dual additive.
Dodecacode
References
- [1]
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [2]
- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
- [3]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [4]
- L. E. Danielsen and M. G. Parker, “On the classification of all self-dual additive codes over GF(4) of length up to 12”, Journal of Combinatorial Theory, Series A 113, 1351 (2006) arXiv:math/0504522 DOI
- [5]
- D. Krotov and P. Sole, “The punctured Dodecacode is uniformly packed”, 2019 IEEE International Symposium on Information Theory (ISIT) 1912 (2019) DOI
- [6]
- J. Kim and V. Pless, Designs, Codes and Cryptography 30, 187 (2003) DOI
- [7]
- A. J. Scott, “Probabilities of Failure for Quantum Error Correction”, Quantum Information Processing 4, 399 (2005) DOI
- [8]
- D. Krotov and P. Sole, “The punctured Dodecacode is uniformly packed”, 2019 IEEE International Symposium on Information Theory (ISIT) 1912 (2019) DOI
Page edit log
- Victor V. Albert (2022-08-11) — most recent
Cite as:
“Dodecacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dodecacode