True Galois-qudit stabilizer code

Description

Also called a linear stabilizer code. A \([[n,k,d]]_{GF(q)}\) 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 \(GF(q)\)-valued 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.

Parent

Child

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 defining each code form a linear subspace.

Zoo code information

Internal code ID: galois_true_stabilizer

Your contribution is welcome!

on github.com (edit & pull request)

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} }
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/tree/main/codes/quantum/qudits_galois/galois_true_stabilizer.yml.