Galois-qudit topological code[13] 

Also known as \(\mathbb{F}_q\)-qudit topological code.


Abelian topological code, such as a 2D surface [1,3] or 2D color [2] code, constructed on lattices of Galois qudits.



  • 2D color code — Galois-qudit 2D color codes reduce to 2D color codes for \(q=2\).
  • Kitaev surface code — The Galois-qudit surface code for \(q=2\) reduces to the surface code.


  • Abelian quantum-double stabilizer code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [4]; see Sec. 5.3 of Ref. [5]. Galois-qudit topological surface and color codes yield Abelian quantum-double codes via this decomposition.


S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007) arXiv:quant-ph/0609070 DOI
P. Sarvepalli, “Topological color codes over higher alphabet”, 2010 IEEE Information Theory Workshop (2010) DOI
I. Andriyanova, D. Maurice, and J.-P. Tillich, “New constructions of CSS codes obtained by moving to higher alphabets”, (2012) arXiv:1202.3338
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
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Zoo Code ID: galois_topological

Cite as:
“Galois-qudit topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_galois_topological, title={Galois-qudit topological code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Galois-qudit topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.