# Stabilizer code over \(GF(q^2)\)[1]

## Description

An \([[n,k,d]]_q\) Galois-qudit stabilizer code constructed from a classical code over \(GF(q^2)\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\).

An \(n\) Galois-qudit Pauli stabilizer can be represented as a length-\(n\) vector over \(GF(q^2)\). The stabilizer commutation condition corresponds to a zero trace-alternating inner product between the corresponding vectors. Stabilizer codes over \(GF(q^2)\) can thus be constructed from classical trace-alternating self-orthogonal additive codes over \(GF(q^2)\) [1]. Hermitian self-orthogonal linear codes over \(GF(q^2)\) are automatically trace-alternating self-orthogonal, and applying this construction to such codes yields a class of true stabilizer codes.

## Parent

- True Galois-qudit stabilizer code — Trace-alternating self-orthogonal linear codes over \(GF(q^2)\) are equivalent to a class of true stabilizer codes [2]. Hermitian self-orthogonal linear codes over \(GF(q^2)\) are automatically trace-alternating self-orthogonal and can be used to construct true stabilizer codes via the stabilizer-over-\(GF(q^2)\) construction ([1], Corr. 19).

## Child

- Stabilizer code over \(GF(4)\) — Stabilizer codes over \(GF(q^2)\) for \(q=2\) are stabilizer codes over \(GF(4)\).

## Cousin

- Dual additive code — The stabilizer commutation condition for stabilizer codes over \(GF(q^2)\) can equivalently be stated in the representation of stabilizers as vectors over \(GF(q^2)\). A pair of \(n\) Galois-qudit stabilizers commute iff the trace-alternating inner product of their their corresponding vectors is zero. Stabilizer codes over \(GF(q^2)\) can thus be constructed from trace-alternating self-orthogonal additive codes over \(GF(q^2)\).

## References

- [1]
- A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
- [2]
- D. Gottesman. Surviving as a quantum computer in a classical world

## Page edit log

- Victor V. Albert (2022-07-22) — most recent
- Victor V. Albert (2022-03-21)

## Cite as:

“Stabilizer code over \(GF(q^2)\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stabilizer_over_gfqsq