Here is a list of codes related to perfect quantum codes.
| Code | Description |
|---|---|
| Hermitian qubit code | A qubit stabilizer code constructed from a Hermitian self-orthogonal linear quaternary code using the Hermitian construction. |
| Modular-qudit CWS code | A CWS code for modular qudits, defined using a modular-qudit cluster state and a set of modular-qudit \(Z\)-type Pauli strings defined by a \(q\)-ary classical code over \(\mathbb{Z}_q\). |
| Modular-qudit GKP code | Modular-qudit analogue of the GKP code. Encodes a qudit into a larger qudit and protects against Pauli shifts up to some maximum value. |
| Modular-qudit shift-resistant code | Monolithic code encoding a qubit into a single modular qudit and protecting against either \(Z\)-type or \(X\)-type modular-qudit Pauli shifts. |
| Perfect code | A type of \(q\)-ary code whose parameters satisfy the Hamming bound with equality. |
| Perfect quantum code | A type of block quantum code whose parameters satisfy the quantum Hamming bound with equality. |
| Quantum data-syndrome (QDS) code | Stabilizer code designed to correct both data qubit errors and syndrome measurement errors simultaneously due to extra redundancy in its stabilizer generators. |
| Quantum twisted code | Hermitian code arising constructed from twisted BCH codes. |
| \([[15, 7, 3]]\) quantum Hamming code | Self-dual quantum Hamming code that admits permutation-based CZ logical gates. The code is constructed using the CSS construction from the \([15,11,3]\) Hamming code and its \([15,4,8]\) dual code. |
| \([[2^r, 2^r-r-2, 3]]\) Gottesman code | A family of pure [1] non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound. |
| \([[5,1,3]]\) Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
| \(q\)-ary Hamming code | Member of an infinite family of perfect linear \(q\)-ary codes with parameters \([(q^r-1)/(q-1),(q^r-1)/(q-1)-r, 3]_q\) for \(r \geq 2\). |
References
- [1]
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006