The object. Label fifteen physical qubits by the nonzero vectors
V* = F_2^4 \ {0000}.
For each of the four coordinate bits, evaluate that bit on every label. The four resulting length-fifteen rows form the X-check matrix H_X. Let 1 denote the all-one row and define the Z-check space by
row(H_Z) = span(H_X, 1)^perp.
This prescription is the whole executable code. The X and Z ranks are four and ten, so fourteen stabilizer constraints leave one logical qubit. A three-point line {a,b,a+b} supports a logical Z; no Pauli of weight one or two is logical. The shipped instance therefore recomputes [[15,1,3]] exactly. The complete rank, quotient, and distance argument is on the punctured-evaluation and logical-distance pages.
The distinctive feature. Put the odd all-one logical-X row above the four even X checks. Every pair and every triple of these five rows has even overlap. That stronger condition—triorthogonality—makes a bitwise T-type phase preserve the code. In the logical convention fixed by the constructor, physical T on all fifteen qubits implements logical T^dagger; physical T^dagger implements logical T. The dagger is derived from codeword weights, not chosen by naming convention; see the transversal-phase derivation.
One sentence. The code is the punctured four-bit Boolean evaluation table with one odd row reserved for the logical qubit: ordinary orthogonality makes it a CSS code, and triple-overlap parity upgrades one diagonal physical layer to a logical non-Clifford phase.
A CSS matrix only asks every X-check support to overlap every Z-check support evenly. That guarantees commutation, but it says almost nothing about a non-Clifford diagonal gate: applying T to a superposition gives each basis word a phase determined by its weight modulo eight. If different words in the same logical basis state acquire different phases, the state leaves the code space even though every Pauli check still commutes.
The fifteen-qubit construction solves both constraints with one Boolean table. A nonzero linear function on F_2^4 is balanced: among the sixteen points it is one on eight, and puncturing 0000 does not change that support. Thus every nonzero X-stabilizer word has weight eight. Distinct coordinate functions overlap on four points; three distinct coordinate functions overlap on two. Those powers of two are exactly the divisibility data needed by a T-type phase.
The price is visible in the same table:
This is why the block appears in two complementary workflows. A distillation factory uses its parity checks to reject likely input magic-state faults. A code-switching protocol moves a logical state between this block, where the T-type phase is transversal, and Steane, where the Clifford symmetry is better. The protocol and evidence ledger keeps those workflows separate.
1. Puncture the zero coordinate. Start from all sixteen vectors of F_2^4, then remove 0000. The remaining labels, in binary integer order 0001, 0010, ..., 1111, are the physical qubits. Each coordinate function vanishes at 0000, so its weight remains eight after puncturing.
2. Reserve the odd row. The four coordinate rows span a dimension-four space L. The all-one row is not in L: every nonzero word in L has weight eight, whereas 1 has weight fifteen. Use L for X stabilizers and reserve 1+L as the logical-X coset.
3. Define Z checks by the exact orthogonal complement. Take
row(H_Z) = (L + span{1})^perp.
Every Z row is orthogonal to every X row, so the CSS checks commute. The space inside the perpendicular has dimension 15-5=10; with rank(H_X)=4, the encoded dimension is 15-4-10=1. The basis printed by constructor.py is a deterministic nullspace basis, but the code depends on its row space rather than on those ten particular generators.
4. Read the two logical sectors. An X operator commuting with all Z checks has support in L + span{1}. Modulo X stabilizers, the only nontrivial coset is 1+L; its words have weight fifteen or seven. A Z operator commuting with all X checks has a support whose labels XOR to zero. The additional all-one orthogonality defining Z stabilizers selects the even-weight part, so the odd part is the logical-Z coset. Three distinct labels a, b, and a+b XOR to zero and give a weight-three logical Z. The sector-distance proof also rules out weights one and two.
5. Strengthen pairwise orthogonality to triple orthogonality. In the five-row matrix G=[1; H_X], an all-one/coordinate pair overlaps on eight qubits, two coordinate rows overlap on four, an all-one/two-coordinate triple overlaps on four, and three coordinate rows overlap on two. Every count is even, so G is triorthogonal.
6. Fix the logical phase. The X-stabilizer superposition for |0_L> contains one word of weight zero and fifteen of weight eight. Its all-one complement, representing |1_L>, contains one word of weight fifteen and fifteen of weight seven. Bitwise physical T therefore gives every |0_L> component phase 1 and every |1_L> component phase exp(-i pi/4). That is logical T^dagger up to a common phase.
7. Separate algebra from protocol. The same X-check equations can be used as an accept/reject test for phase faults in an idealized 15-to-1 block. They do not specify a factory circuit. Likewise, a relation between Steane and Reed–Muller stabilizer groups does not specify a fault-tolerant switching schedule. The decoder/circuit page and protocol ledger name the missing layers explicitly.
L=row(H_X). Then X normalizers are L + span{1}, while Z normalizers are L^perp. Stabilizers quotient out L in the X sector and the even-weight subspace of L^perp in the Z sector. This picture makes one encoded qubit and the d_X=7, d_Z=3 asymmetry explicit. The shared CSS chain-complex dictionary explains how such row-space quotients fit the Atlas convention.[1; H_X] remember more than CSS commutation: pair and triple intersections are even. This is the useful picture for diagonal gates and magic-state distillation. A sign-free symplectic tableau alone cannot determine the accumulated eighth-root phases.The guided incidence tour lets the reader move from four-bit labels to X checks, the orthogonal Z space, the logical quotient, and triorthogonality without confusing graph layout with physical geometry.
Why fifteen physical qubits. There are sixteen inputs to a four-variable Boolean function. Removing the zero input leaves 2^4-1=15 nonzero labels. This puncture turns the all-one row into an odd-weight logical candidate while leaving coordinate-function supports at weight eight.
Why one logical qubit. The four coordinate rows are independent. Adding the independent all-one row gives a dimension-five space, whose orthogonal complement has dimension ten. The CSS stabilizer rank is therefore 4+10=14, so k=15-14=1.
Why total distance three. A single label cannot sum to zero because zero was punctured. Two distinct labels cannot sum to zero over F_2; equality would be required. Thus there is no weight-one or weight-two Z normalizer. For any distinct nonzero a and b, the three labels a, b, and a+b are nonzero and distinct, and their sum is zero. Their odd cardinality keeps the support outside the Z-stabilizer space, producing a logical Z of weight three. Exact CSS enumeration independently returns three.
Why the X distance is seven. Every nonzero linear function is one on eight of the fifteen labels. Complementing it by the all-one logical row gives weight seven; complementing the zero function gives weight fifteen. Hence the nontrivial X coset has minimum weight seven. The scalar distance records the smaller sector, min(7,3)=3, so a bare [[15,1,3]] label hides useful bias information.
Why a T-type phase works. The X-stabilizer words have weight zero modulo eight, whereas every logical-one word has weight seven modulo eight. A bitwise eighth-root phase is therefore constant within each logical basis state but different between them. Pair and triple overlap parity is the local row test for that global weight-divisibility statement.
All structured n, k, d, and CSS claims in instances.yaml are marked verified; this entry uses no bounded distance and no claimed numeric code parameter.
d_Z=3 and d_X=7 in the declared CSS convention. A decoder or biased-noise claim that ignores the sector is underspecified.constructor.py has no size parameter. As of the 2026-08-02 decoder sweep, no source supports an intrinsic asymptotic threshold for this single block. Threshold language requires a growing Reed–Muller family, concatenation level, or full factory family together with its circuit, decoder, observable, and noise normalization.|0_L>,|1_L> basis, physical T^otimes 15 is logical T^dagger, not logical T. Reversing the physical layer reverses the logical gate. A source using the conjugate magic-state convention may name the layer oppositely, so the codeword phase is the invariant statement.Diagonal phase. For a computational-basis word x, bitwise physical T contributes exp(i pi |x|/4). The two logical cosets have weights 0 mod 8 and 7 mod 8, respectively, so the induced diagonal is diag(1, exp(-i pi/4)): logical T^dagger. The inverse physical layer gives logical T. The phase page checks this both by overlap parities and by complete coset enumeration.
Inter-block CNOT. Pair corresponding qubits in two identically ordered blocks. The usual CSS conjugation maps control X checks forward and target Z checks backward while preserving both stabilizer groups, and it induces logical CNOT on the paired quotients. This is one abstract entangling layer; hardware routing and subsequent error correction are additional.
Hadamard through switching. Bitwise H would exchange a rank-four X space with a rank-ten Z space, so it does not preserve this presentation. A fault-tolerant route converts the state to Steane, uses Steane's transversal H, and converts back. The conversion checks, repeated extraction, syndrome-conditioned Pauli frame, and rejected runs belong to the H cost.
15-to-1 distillation. Under the ideal-Clifford, independent input phase-error model, accept exactly the error supports in ker(H_X). Accepted supports of odd weight are logical failures. Their exact enumerator begins with thirty-five weight-three supports, so the conditional output error begins at third order; this is a recomputed block model, not a circuit-level factory threshold. The full enumerator and acceptance denominator are on the protocol ledger.
What hardware has established. A trapped-ion experiment prepared a fifteen-qubit resource, switched it to Steane, and certified the accepted logical magic state. A neutral-atom architecture used 3D fifteen-qubit blocks as teleportation and angle-synthesis resources among other codes and qubit reuse. These are operation-level demonstrations, not increasing-distance Reed–Muller memory scaling; the exact claim boundaries and dates are retained on the evidence ledger.
Q1: Why are the X checks weight eight after the zero point is removed? A: Every nonzero linear function on all sixteen four-bit inputs is balanced with eight ones, and every coordinate function evaluates to zero at 0000. Puncturing that point removes a zero, so the weight stays eight.
Q2: Why does rank(H_Z)=10 rather than eleven? A: Z checks are orthogonal to the four X rows and also to the independent all-one logical-X row. Those five independent constraints on fifteen bits leave a ten-dimensional orthogonal complement.
Q3: Give a logical Z without consulting the printed nullspace basis. A: Choose two distinct nonzero labels a and b; support Z on {a,b,a+b}. The three labels XOR to zero, so the operator commutes with every coordinate X check. Its support has odd size, so it is not orthogonal to the all-one row and therefore is not a Z stabilizer.
Q4: If the code distance is three, why is the minimum logical X weight seven? A: The scalar distance is the minimum across Pauli sectors. The logical-X coset is 1+L, whose words have weights seven or fifteen, while the logical-Z coset contains the three-point lines. Thus d=min(d_X,d_Z)=3 with d_X=7 and d_Z=3.
Q5: Does physical T on every qubit implement logical T? A: Not in this entry's basis. Logical-zero components have weight 0 mod 8, while logical-one components have weight 7 mod 8; physical T therefore produces logical T^dagger. Physical T^dagger produces logical T.
Q6: What extra evidence is needed before the incidence tour becomes a fault-tolerant factory claim? A: At minimum: a preparation and extraction circuit, fault locations and noise normalization, the syndrome/flag history given to a decoder, acceptance and correction rules, and the logical failure observable. The tour verifies stabilizer incidence and phase algebra only.