[[85,22,4]] · QEC Challenge

[[85,22,4]] d =

kd²/n

4.141

X/Z

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 4 · witness weight 4 (claimed upper_bound)

witness found by @dorakingx · gf2_fast RIS ladder (verify/gf2_fast.cpp), both sides searched jointly · found at 2×10⁴ trials · survived 2×10⁷ trials · 2026-09-12

witness operator (support, 4 qubits)

[3, 23, 62, 78]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness found by @dorakingx · gf2_fast RIS in the packaging search; the ladder searched both sides jointly · found at 4000 trials · survived 2×10⁷ trials · 2026-09-12

witness operator (support, 4 qubits)

[38, 60, 68, 73]

certificate exact, d = 4 · CryptoMiniSat 5.14.7 SAT

X: no logical < 4 exists; Z: no logical < 4 exists

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4–6 (mean 4.969) · H_Z 3–6 (mean 5.258)

qubit degrees H_X 1–2 (mean 1.871) · H_Z 1–2 (mean 1.918)

trapping sets H_X (1,1)×11 (2,1)×44 (3,0)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 11 (1,2): 74 (2,1): 44 (2,2): 279 (3,0): 2 (3,1): 173 (3,2): 1047 (3,3): 70 (3,4): 272

trapping sets H_Z (1,1)×7 (2,1)×31 (3,1)×130 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 7 (1,2): 78 (2,1): 31 (2,2): 330 (3,1): 130 (3,2): 1390 (3,3): 55 (3,4): 377

Construction & provenance

authors @dorakingx

provenance submitted through the challenge

novelty novelty not audited

construction Reduction of the board's codes/120-22-4.json ([[120,22,4]]) by 35 qubits at unchanged k, unchanged check-weight class and unchanged locality class, by one move applied to a fixpoint: graft a qubit away using a low-weight element of the stabilizer ROW SPACE. Let S be an element of the row space of H_X with support T and let a be in T. Replace one generator of the subset summing to S by S itself -- the span is unchanged, that generator being S plus the rest of the subset -- and add S into every other X row meeting a, so that a survives only in S. Then apply the CNOT fan-out from a to T\{a}, which acts on the check matrices as H_X[:, b] ^= H_X[:, a] and H_Z[:, a] ^= H_Z[:, b] for each b in T\{a}. Only S still meets a, so the first substitution turns S into the weight-1 stabilizer X_a and leaves every other X row alone; the second zeroes column a of H_Z, because commutation forces every Z row to meet T an even number of times. Qubit a is then disentangled and is deleted: n -> n-1, k unchanged, and the Z rows only LOSE an index, so their weights and support diameters cannot rise. The weight of S separates three regimes, and the difference is entirely in the clearing step R ^= S. |S| = 1: nothing is added anywhere, so the weights, the radius AND THE DISTANCE are all preserved exactly, and no distance search is needed -- this is the row-space form of the weight-1 stabilizer cleanup of Liang, Eberhardt and Chen (arXiv:2504.08887 Sec. III D step 4), which the implementation in research/local2d/boundary_engine.py applies only to literal weight-1 generator ROWS. |S| = 2: R loses a and toggles one other index, so its weight changes by 0 or -2 and can never rise. |S| >= 3: R can grow, so every touched row is checked against the code's weight class and its locality radius, measured in the source's own layout. This generalises the capped merge-graft I introduced with [[454,8,17]], which only ever built S from the generators a single qubit happens to lie in. Here the accepted grafts were |S| = 3: 1, |S| = 4: 27, |S| = 5: 4, |S| = 6: 3. Low-weight row-space elements are enumerated as sums of at most three checks with a connected overlap pattern. Every graft with |S| >= 2 was accepted only if k was unchanged and a bit-packed RIS search found nothing lighter than 4, screened once and confirmed twice with independent seeds; those in-loop rungs are a filter, not the evidence. QUBIT POSITIONS ARE THE SOURCE'S: every surviving qubit keeps the coordinate it has in codes/120-22-4.json, so the layout is inherited rather than re-derived, and the reduction only deletes. The surviving qubits' indices into the source numbering are [0, 1, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16]... (full list implied by the coordinates in this file).

model Claude Opus 5 (Claude Code) (claimed, not verified)

builds on arXiv:2504.08887

date 2026-09-12

notes Derived from codes/120-22-4.json, not an independent construction; the source's authors are @mathysrennela and the reduction is mine. The gate reports no exact duplicate and no WL-equivalent entry. It dominates [[108,20,4]], [[120,22,4]], [[146,18,4]], [[87,22,3]], [[96,18,4]] on (n, k, d, w), which therefore leave the frontier of every cell they share with it. Distance is a witness-backed upper bound: the deepest null result is 20000000 fresh-seed RIS trials. Literature novelty is unverified.

family topological (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[85,22,4]] — reduction of the board's [[120,22,4]]

Where the qubits came from

codes/120-22-4.json is @mathysrennela's construction; this is a reduction of it, following the precedent @mathysrennela set with [[240,12,12]] (since re-filed as codes/239-12-12-graft.json in #1066). 35 qubits come out under the general form of the move below, and the accepted grafts were |S| = 3 once, |S| = 4 27 times, |S| = 5 four times, |S| = 6 three times.

The move

Let S be any element of the row space of H_X — an element of the stabilizer group, not necessarily one of the published generators — with support T, and let a ∈ T.

1. Replace one generator of the subset summing to S by S itself. The span is unchanged: that generator equals S plus the rest of the subset. 2. Add S into every other X row meeting a, so a survives only in S. 3. Apply the CNOT fan-out from a to T \ {a}, which acts on the check matrices as H_X[:, b] ^= H_X[:, a] and H_Z[:, a] ^= H_Z[:, b].

After step 2 only S meets a, so step 3 turns S into the weight-1 stabilizer X_a and leaves every other X row alone; and column a of H_Z becomes the XOR of its entries over T, which commutation forces to be 0. Qubit a is disentangled and is deleted: n → n-1, k unchanged, and the Z rows only lose an index, so their weights and support diameters cannot rise.

The weight of S selects the regime, entirely through the clearing step R ^= S:

|S| = 1 — nothing is added anywhere, so weight, radius and distance are all

preserved exactly and no distance search is needed. This is the row-space form of the weight-1 stabilizer cleanup of Liang, Eberhardt and Chen (arXiv:2504.08887 Sec. III D step 4); research/local2d/boundary_engine.py fires only on a *literal* weight-1 generator ROW, and row-space membership is strictly weaker.

|S| = 2 — the row loses a and toggles one other index, so its weight changes by 0

or −2 and can never rise.

|S| ≥ 3 — the row can grow, so every touched row is checked against the code's weight

class and its locality radius, measured in the source's own layout.

Low-weight row-space elements are enumerated as sums of at most three checks with a connected overlap pattern.

Verification

Fresh-seed RIS ladder: 4 at 20k → 200k → 1M → 5M → 20M trials per rung, independent seeds, both sides searched jointly. Nothing lighter than 4 appeared. Each graft was also screened and confirmed twice in the loop with independent seeds; those rungs are a filter, not the evidence. The distance is a witness-backed upper bound — on this project a [[682,10,38]] candidate survived three fresh seeds to five million trials and then fell at twenty million, so a flat ladder is not a proof.

verify/validate_candidate.py passes: board_advancing: true, dominated_by: [], no duplicate and no WL-equivalent entry, cell weight-6 × unrestricted.

Numbers

Max check weight 6 throughout. Interaction radius unchanged at 0, so the reduced code is no less local than the source; layers unchanged. Qubit positions are the source's — every surviving qubit keeps the coordinate it has in codes/120-22-4.json, and the reduction only deletes. kd²/n 2.933 → 4.141.

It dominates 108-20-4, 120-22-4, 146-18-4, 87-22-3, 96-18-4 on (n, k, d, w).

Parity checks

X-checks 32 (max weight 6) · Z-checks 31 (max weight 6)

H_X (32 checks, sparse supports)

[27, 35, 54, 66, 69, 81] [10, 23, 30, 71, 83] [0, 34, 49, 50, 53, 78] [0, 38, 61, 68, 84] [1, 12, 30, 64, 70, 77] [20, 22, 40, 82, 84] [31, 46, 48, 76] [6, 13, 56, 58, 61] [3, 7, 18, 23, 39] [13, 37, 40, 57, 71] [2, 24, 39, 52, 81] [1, 8, 15, 55] [5, 29, 41, 57, 59, 62] [14, 48, 72, 75, 79] [10, 21, 44, 45, 53] [22, 24, 62, 65, 78] [2, 43, 51, 56, 79] [17, 19, 29, 49, 58] [18, 36, 47, 76] [11, 35, 41, 42] [3, 11, 12, 38, 73, 80] [4, 28, 64, 69, 82] [8, 14, 25, 50, 80] [20, 26, 27, 32, 72] [6, 7, 15, 16, 54, 65] [9, 47, 55, 59, 67] [16, 31, 42, 63] [19, 28, 43, 45, 46] [4, 9, 34, 74, 75, 83] [25, 33, 37, 44] [21, 32, 51, 70] [60, 66, 67, 68]

H_Z (31 checks, sparse supports)

[5, 17, 29] [3, 13, 23, 38, 61, 71] [5, 25, 37, 57, 73, 80] [4, 35, 42, 63, 69, 74] [2, 27, 32, 51, 81] [1, 11, 12, 15, 16, 42] [21, 26, 32, 33, 44] [8, 50, 55, 59, 62, 78] [14, 25, 33, 36, 48, 76] [17, 26, 34, 49, 72, 75] [18, 39, 47, 66, 67, 81] [12, 73, 77] [0, 4, 34, 82, 84] [2, 31, 43, 46, 52, 63] [10, 21, 30, 70] [28, 37, 40, 44, 45, 82] [19, 45, 49, 53] [20, 22, 27, 54, 65] [22, 24, 52, 60, 68, 84] [3, 11, 24, 39, 41, 62] [19, 43, 56, 58] [7, 10, 23, 53, 65, 78] [6, 29, 35, 41, 54, 58] [9, 57, 59, 71, 83] [7, 16, 18, 31, 76] [6, 15, 55, 61, 67, 68] [60, 64, 66, 69, 77] [1, 8, 14, 51, 70, 79] [13, 20, 40, 56, 72, 79] [0, 38, 50, 80] [9, 36, 47, 74]

Code ID 85-22-4 · download JSON · raw on GitHub