\([[5,1,3]]_4\) Galois-qudit CSS code[1]
Description
Five-Galois-qudit CSS code over \(\mathbb{F}_4=\{0,1,\omega,\omega^2\}\) that encodes one logical Galois qudit and corrects a single-qudit error.
Its \(X\)- and \(Z\)-type stabilizer check matrices are \begin{align} H_X&=\begin{pmatrix}1&1&1&1&0\\0&1&\omega&\omega^2&1\end{pmatrix}, \tag*{(1)}\\ H_Z&=\begin{pmatrix}1&1&1&1&0\\0&1&\omega^2&\omega&1\end{pmatrix}~. \tag*{(2)}\end{align} Since the code is a true stabilizer code, multiplication of these rows by \(\omega\) or \(\omega^2\) also yields stabilizers [1].
Cousins
- \([[5,1,3]]\) Five-qubit perfect code— The \([[5,1,3]]_4\) Galois-qudit CSS code is the image of the \([[5,1,3]]\) five-qubit code under the BLT mapping [2; Lemma 1][3; Lemma 1].
- \([5,3,3]_4\) Shortened hexacode— The \([[5,1,3]]_4\) code is obtained from the shortened hexacode [1].
- \([[10,2,3]]\) binarized Galois-qudit code— Binarizing the \([[5,1,3]]_4\) code in the self-dual normal basis \(\{\omega,\omega^2\}\) yields a \([[10,2,3]]\) qubit CSS code [1].
Primary Hierarchy
Parents
The \([[5,1,3]]_4\) code is obtained from the shortened hexacode [1].
The \([[5,1,3]]_4\) code saturates the quantum Singleton bound.
\([[5,1,3]]_4\) Galois-qudit CSS code
References
- [1]
- J. M. Koh, A. Gong, A. C. Diaconu, D. B. Tan, A. A. Geim, M. J. Gullans, N. Y. Yao, M. D. Lukin, and S. Majidy, “Entangling logical qubits without physical operations”, (2026) arXiv:2601.20927
- [2]
- B. W. Reichardt, D. Aasen, and R. Chao, “Fire and ice: Partially fault-tolerant quantum computing with selective state filtering”, (2026) arXiv:2605.15344
- [3]
- S. Bravyi, B. M. Terhal, and B. Leemhuis, “Majorana fermion codes”, New Journal of Physics 12, 083039 (2010) arXiv:1004.3791 DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2026-05-20)
Cite as:
“\([[5,1,3]]_4\) Galois-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/css_5_1_3, arXiv:2606.11484