\([[11,1,5]]_4\) Galois-qudit CSS code[1]
Description
Eleven-Galois-qudit pure CSS code over \(\mathbb{F}_4=\{0,1,\omega,\omega^2\}\) that encodes one logical Galois qudit and corrects up to two single-qudit errors. It is the quantum QR code of length eleven for \(q=4\).
Its \(X\)- and \(Z\)-type stabilizer check matrices are both generator matrices of the self-orthogonal \([11,5,6]_4\) expurgated QR code, which is the cyclic code with generator polynomial \begin{align} (x+1)\,g(x)\quad\text{with}\quad g(x)=x^5+\omega x^4+x^3+x^2+\omega^2 x+1~. \tag*{(1)}\end{align} The Euclidean dual of the expurgated QR code is the \([11,6,5]_4\) QR code with generator polynomial \(g(x)\), whose weight-five codewords furnish minimum-weight logical operators.
Cousins
- \([2m+2,m+1]\) Karlin code— The \([[11,1,5]]_4\) code is constructed from the expurgated QR code obtained by shortening the \([12,6,6]_4\) extended QR code. The binary image of that extended QR code is the \([24,12,8]\) Karlin code [3][2; Ch. 16].
- \([[22,2,6]]\) shortened Golay code— Binarizing the \([[11,1,5]]_4\) code in the self-dual normal basis \(\{\omega,\omega^2\}\) yields the \([[22,2,6]]\) shortened Golay code, raising the distance from five to six.
Primary Hierarchy
References
- [1]
- A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
- [2]
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
- [3]
- P. Gaborit, V. Pless, P. Solé, and O. Atkin, “Type II Codes over F4”, Finite Fields and Their Applications 8, 171 (2002) DOI
Page edit log
- Victor V. Albert (2026-08-28) — most recent
- Shubham P. Jain (2026-08-28)
Cite as:
“\([[11,1,5]]_4\) Galois-qudit CSS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/css_11_1_5, arXiv:2606.11484