[Jump to code hierarchy]

True Galois-qudit stabilizer code[13]

Alternative names: Linear stabilizer code.

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

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

Notes

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

Cousin

Primary Hierarchy

Parents
True Galois-qudit stabilizer code
Children
True Galois-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes.
The code is a non-CSS stabilizer code in general [6].
Galois-qudit BCH codes can be constructed via the CSS construction or the Hermitian construction.
Quantum duadic codes can be constructed via the CSS construction or the Hermitian construction.
Quantum AG codes can be constructed via the Galois-qudit CSS construction or the Galois-qudit Hermitian construction.
Galois-qudit RM codes can be constructed via the Galois-qudit CSS construction or the Galois-qudit Hermitian construction.
Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.
Hermitian codes are true stabilizer codes because they are based on Hermitian self-orthogonal linear (as opposed to additive) codes over \(GF(q^2)\).

References

[1]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[2]
A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “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, S. Yu, H. Fan, and C. H. Oh, “Quantum error-correcting codes over mixed alphabets”, Physical Review A 88, (2013) arXiv:1205.4253 DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: galois_true_stabilizer

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
BibTeX:
@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={https://errorcorrectionzoo.org/c/galois_true_stabilizer} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/galois_true_stabilizer

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/galois_true_stabilizer.yml.