Here is a list of codes related to and generalizing the color code.
Code Description
2D color code Color code defined on a two-dimensional planar graph. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face.
3D color code Color code defined on a four-valent four-colorable tiling of 3D space. Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and types of boundaries (for open surfaces).
Ball color code A color code defined on a \(D\)-dimensional colex. This family includes hypercube color codes (color codes defined on balls constructed from hyperoctahedra) and 3D ball color codes (color codes defined on duals of certain Archimedean solids).
Color code Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs.
Cubic honeycomb color code 3D color code defined on a four-colorable bitruncated cubic honeycomb uniform tiling.
Honeycomb (6.6.6) color code Triangular color code defined on a patch of the 6.6.6 (honeycomb) tiling.
Hyperbolic color code An extension of the color code construction to hyperbolic manifolds. As opposed to there being only three types of uniform three-valent and three-colorable lattice tilings in the 2D Euclidean plane, there is an infinite number of admissible hyperbolic tilings in the 2D hyperbolic plane [1]. Certain double covers of hyperbolic tilings also yield admissible tilings [2]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [3]; see also a construction based on the more general quantum pin codes [4].
Quasi-hyperbolic color code An extension of the color code construction to quasi-hyperbolic manifolds, e.g., a product of a 2D hyperbolic surface and a circle.
Square-octagon (4.8.8) color code Triangular color code defined on a patch of the 4.8.8 (square-octagon) tiling, which itself is obtained by applying a fattening procedure to the square lattice [3].
Stellated color code A non-CSS color code on a lattice patch with a single twist defect at the center of the patch.
Tetrahedral color code 3D color code defined on select tetrahedra of a 3D tiling. Qubits are placed on the vertices, edges, triangles, and in the center of each tetrahedron. The code has both string-like and sheet-like logical operators [5].
Truncated trihexagonal (4.6.12) color code Triangular color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling.
Twist-defect color code A non-CSS extension of the 2D color code whose non-CSS stabilizer generators are associated with twist defects of the associated lattice.
Union-Jack color code Triangular color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack lattice).
XYZ color code Non-CSS variant of the 6.6.6 color code whose generators are \(XZXZXZ\) and \(ZYZYZY\) Pauli strings associated to each hexagonal in the hexagonal (6.6.6) tiling. A further variation called the domain wall color code admits generators of the form \(XXXZZZ\) and \(ZZZXXX\) [6].
\([[15,1,3]]\) quantum Reed-Muller code \([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code.
\([[16,6,4]]\) Tesseract color code A 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube.
\([[2^D,D,2]]\) 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 [7]. Various other concatenations give families with increasing distance (see cousins).
\([[2^r-1,1,3]]\) 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 [8,9]. Each code is a color code defined on a simplex in \(r-1\) dimensions [10,11], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself.
\([[2m,2m-2,2]]\) error-detecting code Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [12; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [13].
\([[4,2,2]]\) Four-qubit code Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error.
\([[6,4,2]]\) error-detecting code Error-detecting six-qubit code with rate \(2/3\) whose codewords are cat/GHZ states. A set of stabilizer generators is \(XXXXXX\) and \(ZZZZZZ\). It is the unique code for its parameters, up to equivalence [13; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [14].
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [15]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.
\([[8,2,2]]\) hyperbolic color code An \([[8,2,2]]\) hyperbolic color code defined on the projective plane.
\([[8,3,2]]\) 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]
E. B. da Silva and W. S. Soares Jr, “Hyperbolic quantum color codes”, (2018) arXiv:1804.06382
[2]
N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1301.6588 DOI
[3]
H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
[4]
C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
[5]
A. Kubica et al., “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018) arXiv:1708.07131 DOI
[6]
K. Tiurev et al., “Domain Wall Color Code”, Physical Review Letters 133, (2024) arXiv:2307.00054 DOI
[7]
D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
[8]
B. Zeng et al., “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
[9]
S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[10]
H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
[11]
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
[12]
N. Rengaswamy et al., “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
[13]
A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[14]
H. Goto, “High-performance fault-tolerant quantum computing with many-hypercube codes”, Science Advances 10, (2024) arXiv:2403.16054 DOI
[15]
B. Shaw et al., “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI