Galois-qudit CSS code[17] 

Also known as Euclidean code.


An \([[n,k,d]]_q \) Galois-qudit true stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Galois-qudit Pauli strings. Codes can be defined from chain complexes over \(GF(q)\) via an extension of qubit CSS-to-homology correspondence to Galois qudits.

The stabilizer generator matrix [8; Def. 2], taking values from \(GF(q)\), is of the form \begin{align} H=\begin{pmatrix}0 & H_{Z}\\ H_{X} & 0 \end{pmatrix} \label{eq:parityg} \tag*{(1)}\end{align} such that the rows of the two blocks must be orthogonal \begin{align} H_X H_Z^T=0~. \label{eq:commG} \tag*{(2)}\end{align} The above condition guarantees that the \(X\)-stabilizer generators, defined in the symplectic representation as rows of \(H_X\), commute with the \(Z\)-stabilizer generators associated with \(H_Z\).

Encoding is based on two related \(q\)-ary linear codes, an \([n,k_X,d_X]_q \) code \(C_X\) and \([n,k_Z,d_Z]_q \) code \(C_Z\), satisfying \(C_X^\perp \subseteq C_Z\). The resulting CSS code has \(k=k_X+k_Z-n\) logical Galois qudits and distance \(d\geq\min\{d_X,d_Z\}\). The \(H_X\) (\(H_Z\)) block of \(H\) (1) is the parity-check matrix of the code \(C_X\) (\(C_Z\)). The requirement \(C_X^\perp \subseteq C_Z\) guarantees (2). Specializing to the case when \(C_Z=[n,k,d]_q\) is dual-containing yields a \([[n,2k-n,\geq d_Z]]_q\) Galois-qudit CSS code with \(C_X = C_Z^\perp\). Basis states for the code are, for \(\gamma \in C_X\), \begin{align} |\gamma + C_Z^\perp \rangle = \frac{1}{\sqrt{|C_Z^\perp|}} \sum_{\eta \in C_Z^\perp} |\gamma + \eta\rangle. \tag*{(3)}\end{align}

Galois-qudit CSS codes can also be understood in terms graphs via the reflexive stabilizer framework, which also allows one to define a code for a given set of Pauli errors [9].


Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.

An \([[n,k,d]]_q\) CSS code can be propagated to an \([[n-2,k,d-1]]_q\) code [10].





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Zoo Code ID: galois_css

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