Here is a list of color codes, with or without defects.

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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 3-colex in a 3-manifold. In the original colex realization, qubits sit on vertices, \(X\)-type stabilizers are attached to 3-cells, and \(Z\)-type stabilizers are attached to faces [1].
Ball code A distance-two color code defined on a colorable \(D\)-ball, equivalently on a \(D\)-colex with boundary [2; Appx. A]. In the morphing construction of Ref. [2], ball codes arise as the child codes associated with the morphed ball-like regions. 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 (typically triangular) patch of the 6.6.6 (honeycomb) tiling. The usual triangular patch has three differently colored boundaries, encodes one logical qubit, and is local-Clifford equivalent to a folded surface/toric code with two smooth and two rough boundaries [3].
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 [4]. Certain double covers of hyperbolic tilings also yield admissible tilings [5]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [1]; see also a construction based on the more general quantum pin codes [6].
Quasi-hyperbolic color code An extension of the color code construction to quasi-hyperbolic 3-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 [1]. An equivalent description uses the Tetrakis square tiling (a.k.a. the Union Jack lattice), which is dual to the 4.8.8 lattice [7]. Among the three semiregular triangular 2D color-code families, the 4.8.8 family uses the fewest physical qubits for a given distance and is the only one of the three with transversal implementations of the full Clifford group [8].
Stellated color code A non-CSS color-code family on a lattice patch with a single central puncture that hosts a twist defect connected to the boundary by a domain wall.
Tetrahedral color code A 3D color code defined on a colored tetrahedron cut from a suitably colored BCC lattice [9]. Qubits are placed on tetrahedra, on the triangles covering the tetrahedron faces, on the edges along the tetrahedron edges, and on the tetrahedron vertices. The code has both string-like and sheet-like logical operators [10].
Truncated trihexagonal (4.6.12) color code 2D color code defined on a (typically triangular) 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. These twists terminate domain walls that permute color labels, Pauli labels, or interchange the two.
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 \(ZZZXXX\) [11].
\([[15,1,3]]\) 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 (hyperbolic 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 [12].
\([[17,1,5]]\) 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 \(n=17\) [13]. It is also a normal self-dual CSS code whose transversal Hadamard acts logically, making it suitable as an inner code for fifth-order magic-state distillation [14].
\([[2^D,D,2]]\) 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. The measured physical bit string can be post-processed into both a logical output string and stabilizer checks, enabling end-of-circuit error detection directly from classical samples [15].
\([[2^r-1,1,3]]\) simplex code Member of a color-code family constructed from a punctured first-order RM\((1,m=r)\) 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 [16,17]. Each code is a color code defined on a simplex in \(r-1\) dimensions [9,18], 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 [19; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [20].
\([[4,1,2]]\) twist-defect code A four-qubit non-CSS stabilizer code that can be interpreted as the smallest triangular color code with \(x\)-, \(y\)-, and \(z\)-type Pauli boundaries [21; Fig. 7], and equivalently as a small twist-defect surface code on a tetrahedron inscribed in a sphere [22]. It is the only non-CSS qubit stabilizer code with parameters \([[4,1,2]]\) [23; ID 8]. The code admits weight-three stabilizer generators \(\{IXXX,YIYY,ZZZI\}\) and weight-two logical Pauli \(X,Y,Z\) operators.
\([[4,2,2]]\) Four-qubit code A four-qubit hyperbolic self-dual CSS stabilizer code that is the smallest two-logical-qubit stabilizer code to detect a single-qubit error. It is unique for its parameters [24; Thm. 8][23; ID 9].
\([[6,2,2]]\) \(C_6\) code Error-detecting normal self-dual CSS code on three qubit pairs that encodes a logical qubit pair and detects any error acting on one pair [25]. In Knill’s \(C_4/C_6\) architecture, this code is used at the second and higher concatenation levels. A choice of check operators used in that construction is \(XIIXXX\), \(XXXIIX\), \(ZIIZZZ\), and \(ZZZIIZ\), with logical operators \(X_L = IXXIII\), \(Z_L = IIZZIZ\), \(X_S = XIXXII\), and \(Z_S = IIIZZI\) [25][23; ID 126].
\([[6,4,2]]\) 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 [20; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [26].
\([[7,1,3]]\) Steane code A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [27][23; ID 226]. 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. In encoded IQP sampling, the final measurement outcomes determine both the logical sample and stabilizer checks, enabling end-of-circuit error detection or postselected decoding directly from the classical samples [15].

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