## Description

An \(((n,K,d))_{GF(q)}\) Galois-qudit code whose logical subspace is the joint eigenspace of commuting Galois-qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a Galois-qudit Pauli string that implements a nontrivial logical operation in the code.

A Galois-qudit stabilizer code encoding an integer number of qudits (\(K=q^k\)) is denoted as \([[n,k]]_{GF(q)}\) or \([[n,k,d]]_{GF(q)}\). This notation differentiates between Galois-qudit and modular-qudit stabilizer codes, although the same notation, \([[n,k,d]]_q\), is usually used for both. Galois-qudit stabilizer codes need not encode an integer number of qudits, with \(K=q^{n-\frac{r}{m}}\), where \(r\) is the number of generators of the stabilizer group, and \(q=p^m\) given prime \(p\) for all Galois qudits. As a result, \([[n,k,d]]\) notation is often used with non-integer \(k=\log_q K\).

## Protection

## Gates

## Parents

- Stabilizer code
- Galois-qudit non-stabilizer code — A non-stabilizer code is also a stabilizer code if its Fourier description \(\mathsf{B}\) is a subgroup of some Gottesman subgroup \(\mathsf{S}\). When \(\mathsf{B}\) is just a subset, the code is explicitly not a stabilizer code.

## Children

## Cousins

- Modular-qudit stabilizer code — Recalling that \(q=p^m\), Galois-qudit stabilizer codes can also be treated as prime-qudit stabilizer codes on \(mn\) qudits, giving \(k=nm-r\) [1]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits.
- Additive \(q\)-ary code — A Galois-qudit stabilizer code is the closest quantum analogue of an additive code over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.

## Zoo code information

## References

- [1]
- Alexei Ashikhmin and Emanuel Knill, “Nonbinary Quantum Stabilizer Codes”. quant-ph/0005008
- [2]
- Avanti Ketkar et al., “Nonbinary stabilizer codes over finite fields”. quant-ph/0508070

## Cite as:

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