Galois-qudit USt code[15] 

Also known as Galois-qudit non-stabilizer code.


A Galois-qubit code whose codespace consists of a direct sum of a Galois-qubit stabilizer codespace and one or more of that stabilizer code's error spaces.

Given a subset \(T\) of coset representatives of \(\mathsf{N}(\mathsf{S})/\mathsf{S}\) of a Galois-qudit stabilizer code \(((n,K))\) with codespace \(\mathsf{C}\) and stabilizer group \(\mathsf{S}\), one can construct the Galois-qudit USt with codespace \begin{align} \mathsf{C}_{\text{USt}}=\bigoplus_{t\in T}t\mathsf{C}~. \tag*{(1)}\end{align} The parameters of the USt are \(((n,K|T|,d))\), where \(|T|\) is the number of chosen coset representatives. A Galois-qudit USt is CSS-like when the underlying stabilizer code is CSS, so the coset representatives from the two classical codes underlying the CSS code.

Union stabilizer codes generalize stabilizer codes by modifying the original stabilizer code projection with elements of a subset \(\mathsf{B}\subset\mathsf{S}\) called the Fourier description [4; Thm. 2.7]. When \(\mathsf{B}\) is a subgroup of \(\mathsf{S}\), then the code reduces to an ordinary stabilizer code.

The \(((n, \lceil\frac{q^n}{n(q^2-1)}\rceil,2))_q\) family of Galois-qudit non-stabilizer codes is constructed in Ref. [4].



  • Union stabilizer (USt) code — Galois-qudit union stabilizer codes reduce to union stabilizer codes for \(q=2\).
  • \(((n,1+n(q-1),2))_q\) union stabilizer code
  • Galois-qudit CWS code — Any Galois-qudit CWS code can be written as a USt whose (\(K=1\)) stabilizer code is the cluster state and whose coset representatives are constructed from the \(q\)-ary classical code.
  • Galois-qudit stabilizer code — A Galois-qudit stabilizer code with stabilizer group \(\mathsf{S}\) can be thought of as a Galois-qudit USt with only the identity coset representative. Conversely, if \(K = q^k\), and if the set of coset representatives of a Galois-qudit USt form a \(q\)-ary linear code, then they can be absorbed into a Galois-qudit stabilizer group that defines the USt.


E. M. Rains et al., “A Nonadditive Quantum Code”, Physical Review Letters 79, 953 (1997) arXiv:quant-ph/9703002 DOI
M. Grassl and T. Beth, “A Note on Non-Additive Quantum Codes”, (1997) arXiv:quant-ph/9703016
V. P. Roychowdhury and F. Vatan, “On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes”, (1997) arXiv:quant-ph/9710031
V. Arvind, P. P. Kurur, and K. R. Parthasarathy, “Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group”, (2002) arXiv:quant-ph/0210097
M. Grassl and M. Rotteler, “Non-additive quantum codes from Goethals and Preparata codes”, 2008 IEEE Information Theory Workshop (2008) arXiv:0801.2144 DOI
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Zoo Code ID: galois_non_stabilizer

Cite as:
“Galois-qudit USt code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_galois_non_stabilizer, title={Galois-qudit USt code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Galois-qudit USt code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.