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

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
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Zoo Code ID: stabilizer_over_gfqsq

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
BibTeX:
@incollection{eczoo_stabilizer_over_gfqsq,
  title={Stabilizer code over \(GF(q^2)\)},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/stabilizer_over_gfqsq}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stabilizer_over_gfqsq

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits_galois/stabilizer/stabilizer_over_gfqsq.yml.