[[158,52,3]] · QEC Challenge

[[158,52,3]] d =

158

kd²/n

2.962

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 17, w_Z = 19 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (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 · 2026-09-15

witness operator (support, 3 qubits)

[15, 17, 18]

d_Z 3 · witness weight 3 (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 · 2026-09-15

witness operator (support, 3 qubits)

[70, 95, 130]

certificate exact, d = 3 · scipy/HiGHS cutoff IP

X: no logical < 3 exists; Z: no logical < 3 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 16–17 (mean 16.44) · H_Z 5–19 (mean 13.012)

qubit degrees H_X 1–5 (mean 2.601) · H_Z 1–44 (mean 6.671)

trapping sets H_X (1,1)×25 (2,1)×375 (3,0)×775 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 25 (1,2): 50 (1,3): 51 (1,4): 27 (1,5): 5 (2,1): 375 (2,2): 703 (2,3): 620 (2,4): 250 (2,5): 60 (2,6): 35 (2,7): 8 (3,0): 775 (3,1): 3503 (3,2): 6381 (3,3): 6068 (3,4): 3814 (3,5): 2533 (3,6): 1760 (3,7): 1060 (3,8): 440 (3,9): 103 (3,10): 12

trapping sets H_Z (1,1)×31 (2,1)×120 (3,0)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 31 (1,2): 61 (1,3): 17 (1,13): 6 (1,14): 5 (1,15): 7 (1,16): 16 (1,17): 8 (1,18): 1 (1,26): 1 (1,27): 2 (1,31): 1 (1,32): 1 (1,44): 1 (2,1): 120 (2,2): 79 (2,3): 7 (2,4): 3 (2,12): 6 (2,13): 143 (2,14): 241 (2,15): 351 (2,16): 570 (2,17): 424 (2,18): 123 (2,19): 15 (2,20): 2 (2,24): 3 (2,25): 15 (2,26): 77 (2,27): 157 (2,28): 117 (2,29): 79 (2,30): 85 (2,31): 111 (2,32): 85 (2,33): 26 (2,34): 4 (2,35): 2 (2,39): 1 (2,40): 5 (2,41): 17 (2,42): 34 (2,43): 45 (2,44): 48 (2,45): 16 (2,46): 5 (2,47): 1 (3,0): 62 (3,1): 120 (3,2): 168 (3,3): 631 (3,4): 550 (3,5): 300 (3,6): 131 (3,7): 26 (3,12): 278 (3,13): 2288 (3,14): 5378 (3,15): 8651 (3,16): 12452 (3,17): 11293 (3,18): 5913 (3,19): 2331 (3,20): 843 (3,21): 239 (3,22): 59 (3,23): 61 (3,24): 383 (3,25): 1467 (3,26): 4082 (3,27): 7220 (3,28): 7932 (3,29): 7315 (3,30): 7159 (3,31): 6490 (3,32): 4327 (3,33): 2012 (3,34): 814 (3,35): 339 (3,36): 98 (3,37): 67 (3,38): 222 (3,39): 732 (3,40): 1734 (3,41): 2801 (3,42): 3328 (3,43): 3137 (3,44): 2315 (3,45): 1252 (3,46): 518 (3,47): 176 (3,48): 67 (3,49): 25 (3,50): 5 (3,51): 1

Construction & provenance

authors @dorakingx

provenance submitted through the challenge

novelty novelty not audited

construction Reduction of the board's codes/170-52-3.json ([[170,52,3]]) by 12 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| = 4: 1, |S| = 6: 1, |S| = 7: 4, |S| = 8: 6. 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 3, 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/170-52-3.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 2026-09-15

notes Derived from codes/170-52-3.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 [[170,52,3]] 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

[[158,52,3]] — reduction of the board's [[170,52,3]]

Where the qubits came from

codes/170-52-3.json is @mathysrennela's construction; this is a reduction of it, following the precedent @mathysrennela set with codes/240-12-12-graft.json. 12 qubits come out under the general form of the move below, and the accepted grafts were |S| = 4 once, |S| = 6 once, |S| = 7 four times, |S| = 8 six 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: 3 at 20k → 200k → 1M → 5M → 20M trials per rung, independent seeds, both sides searched jointly. Nothing lighter than 3 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 19 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/170-52-3.json, and the reduction only deletes. kd²/n 2.753 → 2.962.

It dominates 170-52-3 on (n, k, d, w).

Parity checks

X-checks 25 (max weight 17) · Z-checks 81 (max weight 19)

H_X (25 checks, sparse supports)

[0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 155] [1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141, 151] [2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 122, 132, 142, 152, 155] [3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, 113, 123, 133, 143, 153] [4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 104, 114, 124, 134, 144, 154, 155] [5, 10, 25, 30, 45, 50, 65, 70, 85, 90, 105, 110, 125, 130, 145, 150] [6, 11, 26, 31, 46, 51, 66, 71, 86, 91, 106, 111, 126, 131, 146, 151] [7, 12, 27, 32, 47, 52, 67, 72, 87, 92, 107, 112, 127, 132, 147, 152] [8, 13, 28, 33, 48, 53, 68, 73, 88, 93, 108, 113, 128, 133, 148, 153] [9, 14, 29, 34, 49, 54, 69, 74, 89, 94, 109, 114, 129, 134, 149, 154] [15, 20, 25, 30, 55, 60, 65, 70, 95, 100, 105, 110, 135, 140, 145, 150] [16, 21, 26, 31, 56, 61, 66, 71, 96, 101, 106, 111, 136, 141, 146, 151] [17, 22, 27, 32, 57, 62, 67, 72, 97, 102, 107, 112, 137, 142, 147, 152] [18, 23, 28, 33, 58, 63, 68, 73, 98, 103, 108, 113, 138, 143, 148, 153] [19, 24, 29, 34, 59, 64, 69, 74, 99, 104, 109, 114, 139, 144, 149, 154] [35, 40, 45, 50, 55, 60, 65, 70, 115, 120, 125, 130, 135, 140, 145, 150] [36, 41, 46, 51, 56, 61, 66, 71, 116, 121, 126, 131, 136, 141, 146, 151, 156] [37, 42, 47, 52, 57, 62, 67, 72, 117, 122, 127, 132, 137, 142, 147, 152, 156] [38, 43, 48, 53, 58, 63, 68, 73, 118, 123, 128, 133, 138, 143, 148, 153, 156] [39, 44, 49, 54, 59, 64, 69, 74, 119, 124, 129, 134, 139, 144, 149, 154, 156] [75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150] [76, 81, 86, 91, 96, 101, 106, 111, 116, 121, 126, 131, 136, 141, 146, 151, 157] [77, 82, 87, 92, 97, 102, 107, 112, 117, 122, 127, 132, 137, 142, 147, 152, 157] [78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 148, 153, 157] [79, 84, 89, 94, 99, 104, 109, 114, 119, 124, 129, 134, 139, 144, 149, 154, 157]

H_Z (81 checks, sparse supports)

[0, 3, 4, 37, 38, 42, 43, 155] [1, 2, 3, 4, 15, 16, 18, 20, 21, 23, 155] [2, 3, 5, 8, 9, 10, 13, 14, 155] [5, 7, 9, 10, 12, 14, 155] [10, 12, 14, 55, 57, 59, 65, 67, 69, 155] [5, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23] [15, 18, 19, 40, 43, 44, 60, 63, 64] [15, 17, 19, 20, 22, 24, 155] [20, 22, 24, 35, 37, 39, 55, 57, 59, 155] [20, 23, 24, 37, 38, 40, 42, 44, 60, 63, 64, 155] [25, 27, 29, 35, 37, 39, 65, 67, 69] [2, 3, 10, 13, 14, 25, 28, 29, 40, 43, 44, 60, 63, 64, 155] [5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 27, 28, 29] [30, 32, 34, 35, 37, 39, 65, 67, 69, 155] [2, 3, 10, 13, 14, 30, 33, 34, 37, 38, 40, 42, 44, 60, 63, 64] [5, 6, 8, 10, 12, 14, 21, 22, 23, 24, 31, 32, 33, 34, 155] [36, 37, 38, 39, 156] [15, 16, 18, 20, 21, 23, 41, 42, 43, 44, 155, 156] [45, 47, 49, 55, 57, 59, 65, 67, 69, 110, 112, 114, 145, 147, 149, 155] [2, 3, 10, 13, 14, 35, 38, 39, 45, 48, 49, 155] [5, 6, 8, 10, 12, 14, 46, 47, 48, 49, 155, 156] [50, 52, 54, 55, 57, 59, 65, 67, 69, 110, 112, 114, 145, 147, 149] [2, 3, 10, 13, 14, 35, 37, 39, 42, 43, 50, 53, 54] [5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 23, 51, 52, 53, 54, 156] [35, 38, 39, 40, 43, 44, 55, 58, 59, 60, 63, 64] [15, 16, 18, 20, 22, 24, 56, 57, 58, 59, 155, 156] [35, 37, 39, 55, 57, 59, 60, 62, 64, 110, 112, 114, 145, 147, 149] [35, 37, 39, 40, 42, 44, 155] [21, 22, 23, 24, 61, 62, 63, 64, 156] [35, 37, 39, 110, 112, 114, 145, 147, 149, 155] [2, 3, 10, 13, 14, 35, 38, 39, 40, 43, 44, 60, 63, 64, 65, 68, 69, 155] [5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 66, 67, 68, 69, 156] [2, 3, 10, 13, 14, 35, 37, 39, 40, 42, 44, 60, 63, 64, 70, 73, 74] [5, 6, 8, 10, 12, 14, 21, 22, 23, 24, 71, 72, 73, 74, 155, 156] [70, 72, 74, 75, 77, 79, 145, 147, 149, 155] [10, 13, 14, 75, 78, 79, 90, 93, 94] [76, 77, 78, 79, 157] [70, 72, 74, 80, 82, 84, 145, 147, 149] [10, 13, 14, 37, 38, 42, 43, 80, 83, 84, 90, 93, 94, 155] [15, 16, 18, 20, 21, 23, 81, 82, 83, 84, 155, 157] [55, 57, 59, 65, 67, 69, 70, 72, 74, 85, 87, 89, 145, 147, 149, 155] [2, 3, 85, 88, 89, 90, 93, 94, 155] [5, 6, 8, 10, 12, 14, 86, 87, 88, 89, 155, 157] [55, 57, 59, 65, 67, 69, 70, 72, 74, 90, 92, 94, 145, 147, 149] [2, 3, 37, 38, 42, 43] [5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 23, 91, 92, 93, 94, 157] [35, 37, 39, 55, 57, 59, 70, 72, 74, 95, 97, 99, 145, 147, 149, 155] [10, 13, 14, 40, 43, 44, 60, 63, 64, 90, 93, 94, 95, 98, 99] [15, 16, 18, 20, 22, 24, 96, 97, 98, 99, 155, 157] [35, 37, 39, 55, 57, 59, 70, 72, 74, 100, 102, 104, 145, 147, 149] [10, 13, 14, 37, 38, 40, 42, 44, 60, 63, 64, 90, 93, 94, 100, 103, 104, 155] [21, 22, 23, 24, 101, 102, 103, 104, 157] [35, 37, 39, 65, 67, 69, 70, 72, 74, 105, 107, 109, 145, 147, 149, 155] [2, 3, 40, 43, 44, 60, 63, 64, 90, 93, 94, 105, 108, 109, 155] [5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 106, 107, 108, 109, 157] [2, 3, 37, 38, 40, 42, 44, 60, 63, 64, 90, 93, 94, 110, 113, 114] [5, 6, 8, 10, 12, 14, 21, 22, 23, 24, 111, 112, 113, 114, 155, 157] [70, 72, 74, 110, 112, 114, 115, 117, 119] [10, 13, 14, 35, 38, 39, 90, 93, 94, 115, 118, 119] [116, 117, 118, 119, 156, 157] [70, 72, 74, 110, 112, 114, 120, 122, 124, 155] [10, 13, 14, 35, 37, 39, 42, 43, 90, 93, 94, 120, 123, 124, 155] [15, 16, 18, 20, 21, 23, 121, 122, 123, 124, 155, 156, 157] [55, 57, 59, 65, 67, 69, 70, 72, 74, 110, 112, 114, 125, 127, 129] [2, 3, 35, 38, 39, 90, 93, 94, 125, 128, 129, 155] [5, 6, 8, 10, 12, 14, 126, 127, 128, 129, 155, 156, 157] [55, 57, 59, 65, 67, 69, 70, 72, 74, 110, 112, 114, 130, 132, 134, 155] [2, 3, 35, 37, 39, 42, 43, 90, 93, 94, 130, 133, 134] [5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 23, 131, 132, 133, 134, 156, 157] [35, 37, 39, 55, 57, 59, 70, 72, 74, 110, 112, 114, 135, 137, 139] [10, 13, 14, 35, 38, 39, 40, 43, 44, 60, 63, 64, 90, 93, 94, 135, 138, 139] [15, 16, 18, 20, 22, 24, 136, 137, 138, 139, 155, 156, 157] [35, 37, 39, 55, 57, 59, 70, 72, 74, 110, 112, 114, 140, 142, 144, 155] [10, 13, 14, 35, 37, 39, 40, 42, 44, 60, 63, 64, 90, 93, 94, 140, 143, 144, 155] [21, 22, 23, 24, 141, 142, 143, 144, 156, 157] [35, 37, 39, 65, 67, 69, 70, 72, 74, 110, 112, 114, 145, 147, 149] [2, 3, 35, 38, 39, 40, 43, 44, 60, 63, 64, 90, 93, 94, 145, 148, 149, 155] [5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 146, 147, 148, 149, 156, 157] [35, 37, 39, 65, 67, 69, 70, 72, 74, 110, 112, 114, 150, 152, 154, 155] [2, 3, 35, 37, 39, 40, 42, 44, 60, 63, 64, 90, 93, 94, 150, 153, 154] [5, 6, 8, 10, 12, 14, 21, 22, 23, 24, 151, 152, 153, 154, 155, 156, 157]

Code ID 158-52-3 · download JSON · raw on GitHub