[[141,44,4]] · QEC Challenge

[[141,44,4]] d =

141

kd²/n

4.993

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 16, w_Z = 20 (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)

[72, 73, 75, 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)

[42, 50, 90, 98]

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 9–16 (mean 11.3) · H_Z 9–20 (mean 13.228)

qubit degrees H_X 1–8 (mean 3.206) · H_Z 1–25 (mean 5.348)

trapping sets H_X (1,1)×8 (2,1)×256 (3,1)×280 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 8 (1,2): 32 (1,3): 48 (1,4): 41 (1,5): 8 (1,8): 4 (2,1): 256 (2,2): 384 (2,3): 256 (2,4): 115 (2,5): 108 (2,6): 108 (2,7): 36 (2,8): 32 (2,9): 96 (2,10): 96 (2,11): 32 (3,1): 280 (3,2): 1120 (3,3): 1968 (3,4): 2105 (3,5): 2080 (3,6): 1758 (3,7): 1764 (3,8): 1994 (3,9): 2276 (3,10): 2374 (3,11): 2476 (3,12): 1896 (3,13): 780 (3,14): 112 (3,15): 48 (3,16): 96 (3,17): 48

trapping sets H_Z (1,1)×15 (2,1)×168 (3,1)×105 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 15 (1,2): 40 (1,3): 35 (1,4): 10 (1,8): 15 (1,9): 8 (1,10): 4 (1,15): 3 (1,16): 2 (1,17): 1 (1,22): 1 (1,23): 3 (1,24): 3 (1,25): 1 (2,1): 168 (2,2): 144 (2,3): 40 (2,7): 56 (2,8): 241 (2,9): 328 (2,10): 229 (2,11): 97 (2,12): 23 (2,14): 39 (2,15): 162 (2,16): 202 (2,17): 140 (2,18): 63 (2,19): 14 (2,20): 11 (2,21): 56 (2,22): 147 (2,23): 216 (2,24): 200 (2,25): 117 (2,26): 38 (2,27): 9 (2,28): 10 (2,29): 17 (2,30): 21 (2,31): 16 (2,32): 7 (2,33): 1 (2,35): 1 (2,36): 5 (2,37): 7 (2,38): 4 (2,39): 1 (2,41): 1 (2,42): 4 (2,43): 6 (2,44): 4 (2,45): 1 (3,1): 105 (3,2): 400 (3,3): 817 (3,4): 922 (3,5): 506 (3,6): 244 (3,7): 1338 (3,8): 3780 (3,9): 6390 (3,10): 6265 (3,11): 4178 (3,12): 2055 (3,13): 1179 (3,14): 2438 (3,15): 5654 (3,16): 8225 (3,17): 7944 (3,18): 5397 (3,19): 2829 (3,20): 2469 (3,21): 4832 (3,22): 8265 (3,23): 10051 (3,24): 8989 (3,25): 5949 (3,26): 3075 (3,27): 1730 (3,28): 1716 (3,29): 2066 (3,30): 2090 (3,31): 1581 (3,32): 798 (3,33): 320 (3,34): 374 (3,35): 642 (3,36): 729 (3,37): 522 (3,38): 258 (3,39): 199 (3,40): 336 (3,41): 479 (3,42): 471 (3,43): 328 (3,44): 170 (3,45): 70 (3,46): 29 (3,47): 15 (3,48): 6 (3,49): 1

Construction & provenance

authors @dorakingx

provenance submitted through the challenge

novelty novelty not audited

construction Reduction of the board's codes/148-44-4.json ([[148,44,4]]) by 7 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| = 5: 1, |S| = 6: 2, |S| = 7: 1, |S| = 8: 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/148-44-4.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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]... (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/148-44-4.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 [[148,44,4]] on (n, k, d, w), which therefore leaves the frontier of every cell the two share. Distance is a witness-backed upper bound: the deepest null result is 20000000 fresh-seed RIS trials. Literature novelty is unverified.

family hypergraph product (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight > 8 (computed)

How this code was found

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

[[141,44,4]] — reduction of the board's [[148,44,4]]

Where the qubits came from

codes/148-44-4.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). 7 qubits come out under the general form of the move below, and the accepted grafts were |S| = 5 once, |S| = 6 twice, |S| = 7 once, |S| = 8 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-9plus × unrestricted.

Numbers

Max check weight 20 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/148-44-4.json, and the reduction only deletes. kd²/n 4.757 → 4.993.

It dominates 148-44-4 on (n, k, d, w).

Parity checks

X-checks 40 (max weight 16) · Z-checks 57 (max weight 20)

H_X (40 checks, sparse supports)

[0, 16, 32, 48, 64, 80, 96, 112, 128, 130] [1, 17, 33, 49, 65, 81, 97, 113, 129, 130] [2, 18, 34, 50, 66, 82, 98, 114, 128, 129, 130] [3, 19, 35, 51, 67, 83, 99, 115, 130] [4, 20, 36, 52, 68, 84, 100, 116, 128, 130] [5, 21, 37, 53, 69, 85, 101, 117, 129, 130] [6, 22, 38, 54, 70, 86, 102, 118, 128, 129, 130] [7, 23, 39, 55, 71, 87, 103, 119, 130] [8, 16, 40, 48, 72, 80, 104, 112, 131, 134] [9, 17, 41, 49, 73, 81, 105, 113, 132, 134] [10, 18, 42, 50, 74, 82, 106, 114, 131, 132, 134] [11, 19, 43, 51, 75, 83, 107, 115, 133, 134] [12, 20, 44, 52, 76, 84, 108, 116, 131, 133, 134] [13, 21, 45, 53, 77, 85, 109, 117, 132, 133, 134] [14, 22, 46, 54, 78, 86, 110, 118, 131, 132, 133, 134] [15, 23, 47, 55, 79, 87, 111, 119, 134] [24, 32, 40, 48, 88, 96, 104, 112, 135, 137] [25, 33, 41, 49, 89, 97, 105, 113, 136, 137] [26, 34, 42, 50, 90, 98, 106, 114, 135, 136, 137] [27, 35, 43, 51, 91, 99, 107, 115, 137] [28, 36, 44, 52, 92, 100, 108, 116, 135, 137] [29, 37, 45, 53, 93, 101, 109, 117, 136, 137] [30, 38, 46, 54, 94, 102, 110, 118, 135, 136, 137] [31, 39, 47, 55, 95, 103, 111, 119, 137] [56, 64, 72, 80, 88, 96, 104, 112, 140] [57, 65, 73, 81, 89, 97, 105, 113, 138, 140] [58, 66, 74, 82, 90, 98, 106, 114, 138, 140] [59, 67, 75, 83, 91, 99, 107, 115, 139, 140] [60, 68, 76, 84, 92, 100, 108, 116, 139, 140] [61, 69, 77, 85, 93, 101, 109, 117, 138, 139, 140] [62, 70, 78, 86, 94, 102, 110, 118, 138, 139, 140] [63, 71, 79, 87, 95, 103, 111, 119, 140] [0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120] [1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 97, 105, 113, 121] [2, 10, 18, 26, 34, 42, 50, 58, 66, 74, 82, 90, 98, 106, 114, 122] [3, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83, 91, 99, 107, 115, 123] [4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124] [5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93, 101, 109, 117, 125] [6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126] [7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127]

H_Z (57 checks, sparse supports)

[0, 2, 4, 6, 48, 50, 52, 54, 131, 135] [1, 2, 5, 6, 9, 10, 13, 14, 129, 132] [3, 4, 5, 6, 83, 84, 85, 86, 133, 139] [0, 1, 2, 3, 4, 5, 6, 7, 120, 121, 122, 123, 124, 125, 126, 127, 130] [8, 10, 12, 14, 48, 50, 52, 54, 128, 135] [11, 12, 13, 14, 123, 124, 125, 126, 133] [8, 9, 10, 11, 12, 13, 14, 15, 120, 121, 122, 123, 124, 125, 126, 127, 134] [16, 18, 20, 22, 48, 50, 52, 54, 135] [9, 10, 13, 14, 17, 18, 21, 22, 129] [19, 20, 21, 22, 83, 84, 85, 86, 139] [16, 17, 18, 19, 20, 21, 22, 23, 120, 121, 122, 123, 124, 125, 126, 127, 130, 134] [24, 26, 28, 30, 48, 50, 52, 54, 128, 131] [9, 10, 13, 14, 25, 26, 29, 30, 132, 136] [27, 28, 29, 30, 43, 44, 45, 46, 133] [24, 25, 26, 27, 28, 29, 30, 31, 120, 121, 122, 123, 124, 125, 126, 127, 137] [32, 34, 36, 38, 48, 50, 52, 54, 131] [9, 10, 13, 14, 33, 34, 37, 38, 129, 132, 136] [35, 36, 37, 38, 43, 44, 45, 46, 83, 84, 85, 86, 123, 124, 125, 126, 139] [32, 33, 34, 35, 36, 37, 38, 39, 120, 121, 122, 123, 124, 125, 126, 127, 130, 137] [40, 42, 44, 46, 48, 50, 52, 54, 128] [9, 10, 13, 14, 41, 42, 45, 46, 136] [40, 41, 42, 43, 44, 45, 46, 47, 120, 121, 122, 123, 124, 125, 126, 127, 134, 137] [9, 10, 13, 14, 49, 50, 53, 54, 129, 136] [43, 44, 45, 46, 51, 52, 53, 54, 83, 84, 85, 86, 123, 124, 125, 126, 133, 139] [48, 49, 50, 51, 52, 53, 54, 55, 120, 121, 122, 123, 124, 125, 126, 127, 130, 134, 137] [9, 10, 13, 14, 57, 58, 61, 62, 132, 138] [59, 60, 61, 62, 123, 124, 125, 126, 139] [56, 57, 58, 59, 60, 61, 62, 63, 120, 121, 122, 123, 124, 125, 126, 127, 140] [9, 10, 13, 14, 56, 57, 60, 61, 64, 66, 68, 70, 128, 132, 138] [9, 10, 13, 14, 65, 66, 69, 70, 129, 132, 138] [67, 68, 69, 70, 83, 84, 85, 86, 133] [64, 65, 66, 67, 68, 69, 70, 71, 120, 121, 122, 123, 124, 125, 126, 127, 130, 140] [9, 10, 13, 14, 56, 57, 60, 61, 72, 74, 76, 78, 131, 132, 138] [9, 10, 13, 14, 73, 74, 77, 78, 138] [75, 76, 77, 78, 123, 124, 125, 126, 133, 139] [72, 73, 74, 75, 76, 77, 78, 79, 120, 121, 122, 123, 124, 125, 126, 127, 134, 140] [9, 10, 13, 14, 56, 57, 60, 61, 80, 82, 84, 86, 128, 131, 132, 138] [9, 10, 13, 14, 81, 82, 85, 86, 129, 138] [80, 81, 82, 83, 84, 85, 86, 87, 120, 121, 122, 123, 124, 125, 126, 127, 130, 134, 140] [9, 10, 13, 14, 56, 57, 60, 61, 88, 90, 92, 94, 132, 135, 138] [9, 10, 13, 14, 89, 90, 93, 94, 132, 136, 138] [43, 44, 45, 46, 91, 92, 93, 94, 133, 139] [88, 89, 90, 91, 92, 93, 94, 95, 120, 121, 122, 123, 124, 125, 126, 127, 137, 140] [9, 10, 13, 14, 56, 57, 60, 61, 96, 98, 100, 102, 128, 132, 135, 138] [9, 10, 13, 14, 97, 98, 101, 102, 129, 132, 136, 138] [43, 44, 45, 46, 83, 84, 85, 86, 99, 100, 101, 102, 123, 124, 125, 126] [96, 97, 98, 99, 100, 101, 102, 103, 120, 121, 122, 123, 124, 125, 126, 127, 130, 137, 140] [9, 10, 13, 14, 56, 57, 60, 61, 104, 106, 108, 110, 131, 132, 135, 138] [9, 10, 13, 14, 105, 106, 109, 110, 136, 138] [43, 44, 45, 46, 107, 108, 109, 110, 139] [104, 105, 106, 107, 108, 109, 110, 111, 120, 121, 122, 123, 124, 125, 126, 127, 134, 137, 140] [9, 10, 13, 14, 56, 57, 60, 61, 112, 114, 116, 118, 128, 131, 132, 135, 138] [9, 10, 13, 14, 113, 114, 117, 118, 129, 136, 138] [43, 44, 45, 46, 83, 84, 85, 86, 115, 116, 117, 118, 123, 124, 125, 126, 133] [112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 130, 134, 137, 140] [48, 50, 52, 54, 120, 122, 124, 126, 128, 131, 135] [9, 10, 13, 14, 121, 122, 125, 126, 132]

Code ID 141-44-4 · download JSON · raw on GitHub