QMDS code bounds
Upper and lower bounds for the highest distance in QMDS families
Codes are marked with \(\ast\) if lower and upper bounds meet.
local dimension \(D=3\).
| \(\mathbf{n+k}\) | upper | lower | optimal | Reference |
|---|---|---|---|---|
| \(4\) | \([\![ 4 , 0 , 3 ]\!]_ 3\) | \([\![ 4 , 0 , 3 ]\!]_3\) | \(\ast\) | Hermitean |
| \(6\) | \([\![ 6 , 0 , 4 ]\!]_ 3\) | \([\![ 6 , 0 , 4 ]\!]_3\) | \(\ast\) | Rains~\cite{782103} |
| \(8\) | \([\![ 6 , 2 , 3 ]\!]_ 3\) | \([\![ 6 , 2 , 3 ]\!]_3\) | \(\ast\) | Single-Error |
| \(10\) | \([\![ 10 , 0 , 6 ]\!]_ 3\) | \([\![ 10, 0 , 6 ]\!]_3\) | \(\ast\) | Glynn code~\cite{GLYNN198643} |
| \(12\) | \([\![ 8 , 4 , 3 ]\!]_ 3\) | \([\![ 8 , 4 , 3 ]\!]_3\) | \(\ast\) | Single-Error |
| \(14\) | \([\![ 11 , 3 , 5 ]\!]_ 3\) | \([\![ 10, 4 , 4 ]\!]_3\) | Grassl/Rötteler I | |
| \(16\) | \([\![ 11 , 5 , 4 ]\!]_ 3\) | \([\![ 10, 6 , 3 ]\!]_3\) | Single-Error |
local dimension \(D=4\).
| \(\mathbf{n+k}\) | upper | lower | optimal | Reference | |
|---|---|---|---|---|---|
| \(4\) | \([\![ 4 , 0 , 3 ]\!]_4\) | \([\![ 4 , 0 , 3 ]\!]_4\) | \(\ast\) | Hermitean | |
| \(6\) | \([\![ 6 , 0 , 4 ]\!]_4\) | \([\![ 6 , 0 , 4 ]\!]_4\) | \(\ast\) | Rains~\cite{782103} | |
| \(8\) | \([\![ 8 , 0 , 5]\!]_4\) | \([\![ 6 , 2 , 3 ]\!]_4\) | Single-Error | lower bound optimal for stab codes [BMS24] | |
| \(10\) | \([\![ 10 , 0 , 6 ]\!]_4\) | \([\![ 10 , 0 , 6 ]\!]_4\) | \(\ast\) | Gulliver et al. \cite{4608969} | |
| \(12\) | \([\![ 10 , 2 , 5 ]\!]_4\) | \([\![ 9 , 3 , 4 ]\!]_4\) | Grassl/Rötteler~\cite{7282626} | ||
| \(14\) | \([\![ 14 , 0 , 8 ]\!]_4\) | \([\![ 10 , 4 , 4 ]\!]_4\) | shortening \([\![18,12,4]\!]_4\) | ||
| \(16\) | \([\![ 13 , 3 , 6 ]\!]_4\) | \([\![ 11 , 5 , 4 ]\!]_4\) | Grassl/Rötteler~\cite{7282626} | ||
| \(18\) | \([\![ 18 , 0 , 10]\!]_4\) | \([\![ 12 , 6 , 4 ]\!]_4\) | shortening \([\![18,12,4]\!]_4\) | ||
| \(20\) | \([\![ 16 , 4 , 7 ]\!]_4\) | \([\![ 12 , 8 , 3 ]\!]_4\) | Single-Error | ||
| \(22\) | \([\![ 22 , 0 , 12]\!]_4\) | \([\![ 14 , 8 , 4 ]\!]_4\) | shortening \([\![18,12,4]\!]_4\) | ||
| \(24\) | \([\![ 19 , 5 , 8 ]\!]_4\) | \([\![ 14 , 10, 3 ]\!]_4\) | Single-Error | ||
| \(26\) | \([\![ 23 , 3 , 11 ]\!]_4\) | \([\![ 17 , 9 , 5 ]\!]_4\) | Grassl/Rötteler I | ||
| \(28\) | \([\![ 22 , 6 , 9 ]\!]_4\) | \([\![ 16 , 12, 3 ]\!]_4\) | Single-Error | ||
| \(30\) | \([\![ 26 , 4 , 12 ]\!]_4\) | \([\![ 18 , 12, 4 ]\!]_4\) | Grassl/Rötteler II |
local dimension \(D=5\).
| \(\mathbf{n+k}\) | upper | lower | optimal | Reference |
|---|---|---|---|---|
| \(4\) | \([\![ 4 , 0 , 3 ]\!]_5\) | \([\![ 4 , 0 , 3 ]\!]_5\) | \(\ast\) | Hermitean |
| \(6\) | \([\![ 6 , 0 , 4 ]\!]_5\) | \([\![ 6 , 0 , 4 ]\!]_5\) | \(\ast\) | Rains~\cite{782103} |
| \(8\) | \([\![ 8 , 0 , 5 ]\!]_5\) | \([\![ 8 , 0 , 5 ]\!]_5\) | \(\ast\) | Kim/Lee~\cite{KimLee04} |
| \(10\) | \([\![ 10 , 0 , 6 ]\!]_5\) | \([\![ 10 , 0 , 6 ]\!]_5\) | \(\ast\) | Kim/Lee~\cite{KimLee04} |
| \(12\) | \([\![ 12 , 0 , 7 ]\!]_5\) | \([\![ 10 , 2 , 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(14\) | \([\![ 14 , 0 , 8 ]\!]_5\) | \([\![ 14 , 0 , 8 ]\!]_5\) | \(\ast\) | Ball code~\cite{Ball2019} |
| \(16\) | \([\![ 16 , 0 , 9 ]\!]_5\) | \([\![ 12 , 4 , 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(18\) | \([\![ 18 , 0 , 10 ]\!]_5\) | \([\![ 18 , 0 , 10]\!]_5\) | \(\ast\) | Ball code~\cite{Ball2019} |
| \(20\) | \([\![ 20 , 0 , 11 ]\!]_5\) | \([\![ 14 , 6 , 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(22\) | \([\![ 22 , 0 , 12 ]\!]_5\) | \([\![ 15 , 7 , 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(24\) | \([\![ 24 , 0 , 13 ]\!]_5\) | \([\![ 16 , 8 , 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(26\) | \([\![ 26 , 0 , 14 ]\!]_5\) | \([\![ 17 , 9 , 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(28\) | \([\![ 26 , 2 , 13 ]\!]_5\) | \([\![ 18 , 10, 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(30\) | \([\![ 30 , 0 , 16 ]\!]_5\) | \([\![ 19 , 11, 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(32\) | \([\![ 30 , 2 , 15 ]\!]_5\) | \([\![ 20 , 12, 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(34\) | \([\![ 34 , 0 , 18 ]\!]_5\) | \([\![ 21 , 13, 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(36\) | \([\![ 34 , 2 , 17 ]\!]_5\) | \([\![ 22 , 14, 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(38\) | \([\![ 38 , 0 , 20 ]\!]_5\) | \([\![ 23 , 15, 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(40\) | \([\![ 37 , 3 , 18 ]\!]_5\) | \([\![ 24 , 16, 5 ]\!]_5\) | shortening \([\![26,18,5]\!]_5\) | |
| \(42\) | \([\![ 42 , 0 , 22 ]\!]_5\) | \([\![ 26 , 16, 6 ]\!]_5\) | Grassl/Rötteler I | |
| \(44\) | \([\![ 41 , 3 , 20 ]\!]_5\) | \([\![ 26 , 18, 5 ]\!]_5\) | Grassl/Rötteler I | |
| \(46\) | \([\![ 46 , 0 , 24 ]\!]_5\) | \([\![ 26 , 20, 4 ]\!]_5\) | Grassl/Rötteler I | |
| \(48\) | \([\![ 45 , 3 , 22 ]\!]_5\) | \([\![ 26 , 22, 3 ]\!]_5\) | Single-Error |
References and updates:
Table based on Felix Huber, Markus Grassl, Quantum Codes of Maximal Distance and Highly Entangled Subspaces, Quantum 4, 284 (2020), https://arxiv.org/abs/1907.07733
Updates from the original tables (2020)
- [BMS24] Simeon Ball, Edgar Moreno, Robin Simoens, Stabiliser codes over fields of even order, arXiv:2401.06618
Last updated: 3 March 2026