Description
Three-Galois-qudit CSS code over \(\mathbb{F}_4=\{0,1,\omega,\omega^2\}\) that encodes one logical Galois qudit and detects a single-qudit error.
Its \(X\)- and \(Z\)-type stabilizer check matrices are both \begin{align} H_X=H_Z=\begin{pmatrix}1&\omega&\omega^2\end{pmatrix}~. \tag*{(1)}\end{align} Since the code is \(\mathbb{F}_4\)-linear, multiplication of this row by \(\omega\) or \(\omega^2\) also yields stabilizers [2].
The code is the smallest member of a family of \(\mathbb{F}_4\)-linear CSS quantum QR codes used in the binarization-and-concatenation construction of phantom qubit codes [2].
Protection
Detects a single Galois-qudit error. It is a quantum MDS code, saturating the quantum Singleton bound.Cousins
- \([4,2,3]_4\) RS\(_4\) code— Puncturing the \([4,2,3]_4\) RS\(_4\) code yields a \([3,2,2]_4\) code whose dual is generated by \((1,\omega,\omega^2)\), the parity-check row used for both \(X\)- and \(Z\)-type stabilizers of the \([[3,1,2]]_4\) code.
- \([[6,2,2]]\) \(C_6\) code— Binarizing the \([[3,1,2]]_4\) code in the self-dual normal basis \(\{\omega,\omega^2\}\) yields a \([[6,2,2]]\) qubit CSS code equivalent to the \(C_6\) code after the qubit relabeling \((2\,3\,4\,5\,6)\) [2].
- \([[12,2,4]]\) carbon code— Binarizing this code and concatenating each qubit pair with the \([[4,2,2]]\) code yields the \([[12,2,4]]\) carbon code [2].
- Binarized-and-concatenated (B&C) phantom code— Binarizing the \([[3,1,2]]_4\) Galois-qudit CSS code and concatenating each qubit pair with the \([[4,2,2]]\) code yields the \([[12,2,4]]\) carbon code [2].
Primary Hierarchy
Parents
The \([[3,1,2]]_4\) code is the smallest \(\mathbb{F}_4\)-linear CSS quantum QR code in the construction of Ref. [2].
The \([[3,1,2]]_4\) code saturates the quantum Singleton bound.
\([[3,1,2]]_4\) Galois-qudit code
References
- [1]
- F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H. N. Ward, “Self-dual codes over GF(4)”, Journal of Combinatorial Theory, Series A 25, 288 (1978) DOI
- [2]
- 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:
“\([[3,1,2]]_4\) Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/galois_3_1_2