[[119,37,4]] · QEC Challenge

[[119,37,4]] d ≤

119

kd²/n

4.975

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · w_X = 6, w_Z = 5 (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 · 2026-10-02

witness operator (support, 4 qubits)

[1, 14, 24, 31]

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 · 2026-10-02

witness operator (support, 4 qubits)

[0, 18, 50, 70]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 10 · H_Z 12 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 4–5 (mean 4.955)

qubit degrees H_X 1–2 (mean 1.916) · H_Z 1–2 (mean 1.832)

trapping sets H_X (1,1)×10 (2,1)×50 (3,1)×250 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 10 (1,2): 109 (2,1): 50 (2,2): 520 (3,1): 250 (3,2): 2475 (3,3): 100 (3,4): 660

trapping sets H_Z (1,1)×20 (2,1)×79 (3,1)×313 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 20 (1,2): 99 (2,1): 79 (2,2): 353 (3,1): 313 (3,2): 1243 (3,3): 117 (3,4): 311

Construction & provenance

authors @dorakingx

provenance submitted through the challenge

novelty novelty not audited

construction Reduction of the board's codes/120-37-4.json ([[120,37,4]]) by 1 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 . 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/120-37-4.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 2026-10-02

notes Derived from codes/120-37-4.json, not an independent construction; the source's authors are @vprusso and the reduction is mine. The gate reports no exact duplicate and no WL-equivalent entry. It dominates [[119,35,3]], [[119,36,3]], [[119,37,3]], [[120,22,4]], [[120,35,4]], [[120,36,2]], [[120,37,4]], [[125,25,4]], [[126,36,2]], [[144,36,4]], [[146,18,4]] 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 check-deletion (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

[[119,37,4]] — reduction of the board's [[120,37,4]]

Where the qubits came from

codes/120-37-4.json is @vprusso's construction; this is a reduction of it, following the precedent @mathysrennela set on this board of submitting a reduction of someone else's entry as a contribution in its own right. 1 qubits come out under the general form of the move below, and the accepted grafts were .

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/120-37-4.json, and the reduction only deletes. kd²/n 4.933 → 4.975.

It dominates 119-35-3, 119-36-3, 119-37-3, 120-22-4, 120-35-4, 120-36-2, 120-37-4, 125-25-4, 126-36-2, 144-36-4, 146-18-4 on (n, k, d, w).

Parity checks

X-checks 38 (max weight 6) · Z-checks 44 (max weight 5)

H_X (38 checks, sparse supports)

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

H_Z (44 checks, sparse supports)

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

Code ID 119-37-4 · download JSON · raw on GitHub