## Description

A \([[n,k,d]]_q\) stabilizer code whose stabilizer's Galois 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.

## Protection

## Notes

## Parent

## Children

- Qubit stabilizer code — True Galois-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes.
- \([[5,1,3]]_q\) Galois-qudit code
- \([[6,2,3]]_{q}\) code — The code is a non-CSS stabilizer code in general [6].
- \([[7,3,3]]_{q}\) code
- Galois-qudit BCH code — Galois-qudit BCH codes can be constructed via the CSS construction or the Hermitian construction.
- Galois-qudit CSS code — Galois-qudit CSS codes are true stabilizer codes [3].
- Quantum duadic code — Quantum duadic codes can be constructed via the CSS construction or the Hermitian construction.
- Galois-qudit quantum RM code — Galois-qudit RM codes can be constructed via the CSS construction or the Hermitian construction.
- Galois-qudit GRS code — Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.
- Hermitian Galois-qudit code — Hermitian codes are true stabilizer codes because they are based on Hermitian self-orthogonal linear (as opposed to additive) codes over \(GF(q^2)\).

## Cousin

- 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 Galois symplectic representation of the stabilizers form a linear subspace.

## References

- [1]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [2]
- A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
- [3]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [4]
- A. Klappenecker, “Algebraic quantum coding theory”, Quantum Error Correction 307 (2013) DOI
- [5]
- M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [6]
- Z. Wang et al., “Quantum error-correcting codes over mixed alphabets”, Physical Review A 88, (2013) arXiv:1205.4253 DOI

## Page edit log

- Victor V. Albert (2022-07-22) — most recent
- Daniel Gottesman (2022-02-17)
- Victor V. Albert (2022-02-17)

## Cite as:

“True Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_true_stabilizer