True Galois-qudit stabilizer code[13] 


Also called a linear stabilizer code. A \([[n,k,d]]_q\) stabilizer code whose stabilizer's symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(GF(q)\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only.

The number of generators \(r\) for a true stabilizer code is a multiple of \(m\) (recall that \(q=p^m\) for Galois qudits). As a result, the number \(k=n-r/m\) of logical qudits is an integer.

Each code can be represented by a stabilizer generator matrix \(H=(A|B)\), where each row \((a|b)\) is the Galois symplectic representation of a stabilizer generator.


Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits.


See Ref. [4,5] for introductions to various stabilizer code constructions.




  • Linear \(q\)-ary code — A true Galois-qudit stabilizer code is the closest quantum analogue of a linear code over \(GF(q)\) because the \(q\)-ary vectors corresponding to the symplectic representation of the stabilizers form a linear subspace.


A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
D. Gottesman. Surviving as a quantum computer in a classical world
A. Klappenecker, “Algebraic quantum coding theory”, Quantum Error Correction 307 (2013) DOI
M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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Zoo Code ID: galois_true_stabilizer

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