Here is a list of codes related to quantum Reed-Muller codes.
Code Description
Galois-qudit quantum RM code True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [1; Sec. 4.2].
Prime-qudit RM code Modular-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes or their duals via the modular-qudit CSS construction. An odd-prime-qudit CSS code family constructed from first-order punctured GRM codes transversally implements a diagonal gate at any level of the qudit Clifford hierarchy [2].
Quantum Reed-Muller code A CSS code formed from a classical Reed-Muller (RM) code or its punctured/shortened versions. Such codes often admit transversal logical gates in the Clifford hierarchy.
\([[15, 7, 3]]\) 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.
\([[15,1,3]]\) quantum Reed-Muller code \([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code.
\([[2^D,D,2]]\) hypercube quantum code Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. It can be generalized to a \([[4^D,D,4]]\) error-correcting family [3]. Various other concatenations give families with increasing distance (see cousins).
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code Member of a family of self-dual CCS codes constructed from \([2^r-1,2^r-r-1,3]=C_X=C_Z\) Hamming codes and their duals the simplex codes. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix for a simplex code. The weight of each stabilizer generator is \(2^{r-1}\).
\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code A family of CSS codes extending quantum Hamming codes to prime qudits of dimension \(p\) by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [4].
\([[2^r-1,1,3]]\) simplex code Member of color-code code family constructed with a punctured first-order RM\((1,m=r)\) \([2^r-1,r+1,2^{r-1}-1]\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [5,6]. Each code is a color code defined on a simplex in \(r-1\) dimensions [7,8], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself.
\([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code Member of CSS code family constructed with a punctured self-dual RM \([2^r-1,2^{r-1},\sqrt{2}^{r-1}-1]\) code and its even subcode for \(r \geq 2\).
\([[4,2,2]]\) Four-qubit code Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error.
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [9]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.
\([[8,3,2]]\) CSS code Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal CCZ gate.

References

[1]
C.-Y. Lai and C.-C. Lu, “A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices”, IEEE Transactions on Information Theory 57, 7163 (2011) arXiv:0712.0103 DOI
[2]
E. T. Campbell, H. Anwar, and D. E. Browne, “Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes”, Physical Review X 2, (2012) arXiv:1205.3104 DOI
[3]
D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
[4]
A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
[5]
B. Zeng, H. Chung, A. W. Cross, and I. L. Chuang, “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
[6]
S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[7]
H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
[8]
B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
[9]
B. Shaw, M. M. Wilde, O. Oreshkov, I. Kremsky, and D. A. Lidar, “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI