Here is a list of codes related to perfect quantum codes.

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Code Description
Five-qubit perfect code Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error.
Hermitian qubit code An \([[n,k,d]]\) stabilizer code constructed from a Hermitian self-orthogonal linear quaternary code using the \(GF(4)\) representation.
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 non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound.
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Error correction zoo by Victor V. Albert, Philippe Faist, and many contributors. This work is licensed under a CC-BY-SA License. See how to contribute.