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 \(GF(4)=\{0,1,\omega,\bar{\omega}\}\) [2; Sec. 2.4.8]. Another generator matrix can be found in [3; Ex. 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]. This code is uniformly packed [6].

Parent

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 [7].
  • \([[11,1,5]]\) 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 [8]. A pure \([[10,1,4]]\) quantum code can be obtained from the doubly punctured dodecacode [8]. These codes are not obtained from the Hermitian construction since none of the classical codes are linear.

References

[1]
A. R. Calderbank et al., “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) (2019) DOI
[6]
D. Krotov and P. Sole, “The punctured Dodecacode is uniformly packed”, 2019 IEEE International Symposium on Information Theory (ISIT) (2019) DOI
[7]
J. Kim and V. Pless, Designs, Codes and Cryptography 30, 187 (2003) DOI
[8]
A. J. Scott, “Probabilities of Failure for Quantum Error Correction”, Quantum Information Processing 4, 399 (2005) DOI
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Zoo Code ID: dodecacode

Cite as:
“Dodecacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dodecacode
BibTeX:
@incollection{eczoo_dodecacode, title={Dodecacode}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dodecacode} }
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Cite as:

“Dodecacode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dodecacode

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