Here is a list of codes related to and generalizing the color code.

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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 code A distance-two “morphed” color code defined on a \(D\)-dimensional colex [1; Appx. A]. This family includes hypercube codes (defined on balls constructed from hyperoctahedra) and 3D ball 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 2D 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 [2]. Certain double covers of hyperbolic tilings also yield admissible tilings [3]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [4]; see also a construction based on the more general quantum pin codes [5].
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 2D 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 [4].
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 [6].
Truncated trihexagonal (4.6.12) color code 2D 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 2D 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 \(ZZZXXX\) [7].
\([[15,1,3]]\) quantum RM code A \([[15,1,3]]\) quantum Reed-Muller code that is most easily thought of as a tetrahedral 3D color code.
\([[16,6,4]]\) Tesseract color code A (self-dual CSS) 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube. A \([[16,4,2,4]]\) tesseract subsystem code can be obtained from this code by using two logical qubits as gauge qubits [8].
\([[17,1,5]]\) 4.8.8 color code Seventeen-qubit doubly even 2D color code that admits a transversal implementation of the logical Clifford group. The smallest distance-five CSS code has \(n=17\) [9].
\([[2^D,D,2]]\) 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 [10]. 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 [11,12]. Each code is a color code defined on a simplex in \(r-1\) dimensions [13,14], 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 [15; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [16].
\([[4,2,2]]\) Four-qubit code Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error.
\([[6,2,2]]\) \(C_6\) code Error-detecting self-dual CSS code used in concatenation schemes for fault-tolerant quantum computation. A set of stabilizer generators is \(IIXXXX\) and \(XXIIXX\), together with the same two \(Z\)-type generators.
\([[6,4,2]]\) 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 [16; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [17].
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [18]. 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.
\([[8,3,2]]\) Smallest interesting color code Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal \(CCZ\) gate.

References

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[8]
B. W. Reichardt et al., “Demonstration of quantum computation and error correction with a tesseract code”, (2024) arXiv:2409.04628
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S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
[13]
H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
[14]
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
[15]
N. Rengaswamy, R. Calderbank, H. D. Pfister, and S. Kadhe, “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) 791 (2018) arXiv:1803.06987 DOI
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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
[17]
H. Goto, “High-performance fault-tolerant quantum computing with many-hypercube codes”, Science Advances 10, (2024) arXiv:2403.16054 DOI
[18]
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