\([[5,1,3]]_4\) 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}, \qquad H_Z=\begin{pmatrix}1&1&1&1&0\\0&1&\omega^2&\omega&1\end{pmatrix}~. \tag*{(1)}\end{align} Since the code is \(\mathbb{F}_4\)-linear, multiplication of these rows by \(\omega\) or \(\omega^2\) also yields stabilizers [1].
Cousins
- \([6,3,4]_4\) Hexacode— The \([[5,1,3]]_4\) code is obtained from the \([6,3,4]_4\) hexacode, the corresponding extended quaternary QR code, by puncturing the extended coordinate in the CSS quantum-QR construction [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].
- Binarized-and-concatenated (B&C) phantom code— Binarizing this code and concatenating each qubit pair with the \([[4,2,2]]\) code yields a \([[20,2,6]]\) B&C phantom code admitting fold-diagonal logical \(SS\) gates [1].
Primary Hierarchy
Parents
The \([[5,1,3]]_4\) code is a Galois-qudit CSS code over \(\mathbb{F}_4\) [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
Page edit log
- Victor V. Albert (2026-05-20) — most recent
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