This entry fixes the binary stabilizer row space generated, in order, by XZZX_, _XZZX, X_XZZ, and ZX_XZ. It also fixes logical representatives XXXXX and ZZZZZ. Qubit permutations, stabilizer row operations, local Clifford conjugations, and logical-basis changes may produce equivalent codes, but executable claims are always interpreted through this representative.
Rains proves uniqueness of the one-qubit-in-five single-error-correcting code up to equivalence (primary source). “Up to equivalence” is exactly why representation dictionaries are needed; it is not license to identify every nearby family with the same file.
Quantum perfect. The zero syndrome plus all fifteen nonzero syndromes are occupied once by the radius-one Pauli error set. The block is one perfect-code member.
Quantum MDS. Its verified parameters obey n-k=2(d-1). The block is one binary quantum-MDS member, not the class of every code saturating that bound.
Frobenius. Dutta and Kurur include the Laflamme example in a cyclic Frobenius construction (primary source). Other lengths, primes, and linearity cases remain different members.
The card points outward with special_case_of in all three cases. Reversing an arrow would wrongly collapse a broad family into the fixed block.
Binary symplectic. Pauli strings become [x|z] rows; commutation, syndrome, rank, and logical quotient become GF(2) operations.
Additive GF(4). Each (x_i,z_i) pair becomes one field symbol. This compresses the binary symplectic data and is useful for classifying additive codes. Physical carriers remain qubits.
Graph code. Schlingemann and Werner construct an encoding from a graph and finite abelian group, including a fivefold one-error-correcting example (primary source). The encoding input is part of the graph-code object.
CWS. Cross and collaborators separate a graph state from classical word operators (primary source). The graph state alone is one state, whereas this code is a two-dimensional subspace.
Encoder-respecting graph. Khesin and collaborators give an efficiently computable tableau-to-graph bijection that keeps encoder interfaces visible (primary source). Transferred logical claims still require conversion back to the tableau convention.
Qudit fivefold codes. Changing local dimension changes the Pauli group, syndrome alphabet, and parameter theorem. The present constructor has no q parameter.
Approximate amplitude-damping codes. These target a restricted physical channel and may outperform exact Pauli correction there. Their approximate Knill-Laflamme contract is not the exact block's identity.
Concatenated codes. Choosing an outer code creates a growing construction whose thresholds and effective channels depend on levels and recovery.
Floquet or pair-measurement codes. A periodic measurement schedule defines the protected subspace through time. A static stabilizer block can inspire a schedule without becoming that dynamical code.
Twisted XZZX and hyperbicycle families. Reusing an XZZX motif inside a larger geometry or polynomial construction does not preserve the exact row space, dimension, or decoder.
Holographic networks. Using the five-qubit encoding isometry as a perfect tensor builds a different many-tensor code. The constituent isometry and assembled network are not parent and child under the current Atlas grammar.
Majorana representations. The pinned source relies on a private communication for a code-specific locality claim. As of 2026-08-18 no public primary source checked specifies enough fermionic parity and locality conventions to include that view honestly.
The pentagon viewer exports five cyclic positions and four mixed-Pauli check objects. It teaches translation and support incidence. Each check touches four positions, so the drawing does not assert nearest-neighbor measurement, two-dimensional locality, a graph-state Hamiltonian, or a Floquet schedule. Those extra structures need separate executable data.