Question. How do fifteen four-bit labels determine the complete stabilizer space, rather than merely suggest a Reed–Muller analogy?
Everything below uses the constructor's fixed order: qubit q carries the four-bit binary expansion of q+1, from 0001 through 1111, with the most-significant bit first.
Let V=F_2^4 and V*=V\{0}. For u in F_2^4, define the length-fifteen word
ell_u = (u dot v)_(v in V*).
The map u -> ell_u is linear and injective. If ell_u vanished for every nonzero v, then it would vanish on the four standard basis vectors and force u=0. Its image
L = {ell_u : u in F_2^4}
therefore has dimension four. The constructor uses the four coordinate functions as an ordered basis:
000000011111111
000111100001111
011001100110011
101010101010101Every nonzero ell_u has weight eight. On all sixteen inputs a nonzero linear functional has an eight-dimensional zero half and an eight-dimensional one half; its value at the removed input 0000 is zero, so puncturing does not change the weight.
This Boolean-evaluation starting point is the route given by Steane's quantum Reed–Muller construction. The Atlas fixes an explicit basis and coordinate order so every later statement is executable.
Let 1 be the length-fifteen all-one word. It is not in L: the zero word in L has weight zero and every nonzero word has weight eight, while 1 has weight fifteen. Hence
G = L + span{1}
has dimension five. The code uses L as its X-stabilizer space and reserves the other coset 1+L for logical X. This choice is essential. If the odd row were also declared an X stabilizer, the encoded dimension and diagonal phase action would both change.
Define
S_Z = G^perp = {z : z dot ell_u = 0 for all u, and z dot 1 = 0}.
The constructor obtains an ordered basis with gf2.nullspace(vstack([X_CHECKS, LOGICAL_X])). Its ten output rows are a basis of S_Z, not ten individually canonical geometric faces. Any invertible row recombination gives the same Z-stabilizer group.
Because S_Z is orthogonal to L,
H_X H_Z^T = 0.
Thus every X generator commutes with every Z generator. Same-type generators already commute, so CheckMatrix.from_css(H_X,H_Z) is a valid stabilizer matrix. The verifier checks this product and calls validate() on the full symplectic array.
The extra condition z dot 1=0 says every Z stabilizer has even support. It also distinguishes Z stabilizers from the odd logical-Z coset used on the distance page.
The X rank is four. Since G has dimension five in a fifteen-dimensional ambient space, S_Z=G^perp has dimension ten. X-only and Z-only stabilizer rows occupy disjoint symplectic halves, so their ranks add:
rank(S)=4+10=14.
A rank-fourteen stabilizer on fifteen qubits encodes
k=15-14=1.
This count is independent of the particular nullspace basis printed in constructor.py. It depends only on the evaluation space L and the reserved all-one row.
The repository's [[15,7,3]] quantum Hamming entry uses the same four coordinate rows for both X and Z checks. Its four-dimensional Z space is a subspace of this code's ten-dimensional S_Z. Adding six independent Z constraints fixes six of the Hamming block's seven logical degrees of freedom and leaves this one-qubit block. The companion entry verifies that literal row-space containment in the shared qubit order.
This relation is gauge-out/dimension reduction, not two names for the same stabilizer code. The ranks 8 and 14 already rule out that conflation.
The evaluation points, matrices, ranks, commutation, and encoded dimension are recomputed by pedagogy/reproduce.py. The historical naming and the fact that this block is the punctured four-variable member of the broader quantum Reed–Muller lineage remain cited to the primary literature. The tetrahedral 3D color-code realization is a separate curated dictionary; the present proof uses no unexported geometric assumption.