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[[192,43,12]] d ≤
n
192
k
43
d
12
kd²/n
32.25
w
12
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 12, w_Z = 12 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[4, 19, 32, 61, 71, 91, 107, 117, 130, 159, 174, 183]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[1, 23, 36, 43, 49, 63, 67, 78, 104, 126, 139, 152]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 12 · H_Z 12
qubit degrees H_X 4–5 (mean 4.75) · H_Z 5
trapping sets H_X (1,4)×48 (2,3)×16 (3,4)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 48 (1,5): 144 (2,3): 16 (2,4): 58 (2,5): 276 (2,6): 854 (2,7): 1016 (2,8): 1736 (3,4): 100 (3,5): 482 (3,6): 2222 (3,7): 6602 (3,8): 16576 (3,9): 31496 (3,10): 26964 (3,11): 31338 (3,12): 1248 (3,13): 1300
trapping sets H_Z (1,5)×192 (2,4)×48 (3,5)×384 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,5): 192 (2,4): 48 (2,6): 1072 (2,8): 2992 (3,5): 384 (3,7): 6848 (3,9): 47488 (3,11): 69664 (3,13): 2944

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Greedy check-deletion chain from board code codes/192-40-12.json (GALA code of arXiv:2608.07431 Table S5; the paper certifies d = 12 exactly). Deleted 4 X-checks [0, 1, 8, 9] over 4 greedy steps, each priced by the check-deletion theorem (unpublished theorem by @mathysrennela, claim 12.1). k' = 40 + 4 = 43 exact by rank; the deleted X rows weigh exactly 12 and are the X-side witnesses (term 2); the Z witness was re-confirmed at weight 12 by the trusted gate.
model GLM 5.3 Flash (claimed, not verified)
builds on https://arxiv.org/abs/2608.07431
date 2026-09-09
notes Derived code, honest provenance: check-deletion child of its source entry codes/192-40-12.json (k 40 -> 43 at equal n, distance tier, and weight class; different check sets, not equivalent to the source under any code symmetry). Deleted 4 X-checks [0, 1, 8, 9]; the deleted X rows are the embedded per-side X witnesses, the source's Z witness is inherited unchanged (term-1 of the check-deletion theorem, an unpublished theorem by @mathysrennela). Parameter novelty vs the literature unverified.
family other (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

[[192,43,12]] — check-deletion chain of codes/192-40-12.json

Direction & hypothesis

Check-deletion move (an unpublished check-deletion theorem by @mathysrennela, PROVEN by the author; numbered claim 12.1 in the author's taxonomy): deleting r independent checks gives k' = k + r exact and `d' = min(d, min deleted-check weight, min exposed-logical weight)`. The board-wide wave-1 census (529 sources, 39,818 moves, 348 survivors at 31 distinct points) promoted [[700,212,28]] (PR #934, merged). Wave 2 asked whether *greedy chains* keep going: delete one check per step, re-price, repeat. Hypothesis for this source: its source is the GALA code of arXiv:2608.07431 (paper-certified d = 12 exact) with heavy X-checks carrying k-slack at the distance tier.

What was searched

Greedy chain (driver kept as local staging; the method here is complete): at each step, enumerate candidate deletions, price each by the theorem's free bound (term 2: the deleted row's weight), prune on combined-Tanner connectivity (issue #921 semantics, pre-filtered as a gate-in-waiting), qubit coverage (no qubit unchecked on either side), and board potential; delete the best candidate; recompute k exactly by GF(2) rank; repeat to a depth cap of 10. Endpoint screening: 2500 RIS trials/side, then the trusted gate (validate_candidate: verify + 8000-trial refutation + dedup + novelty).

Evidence trail

  • k: 40 -> 43, exact by GF(2) rank arithmetic (never bookkeeping).
  • Claim: d <= 12 per side, upper_bound confidence. The deleted X-checks [0, 1, 8, 9] weigh exactly 12 each and are the X-side witnesses (term 2). The Z witness was re-confirmed at weight 12 by the trusted gate, consistent with the source's paper-exact d = 12.
  • Trusted verifier passed locally (verify/qldpc_verify.py codes/192-43-12.json,

    • earned_distance block).

  • Board position: strictly dominates its source entry codes/192-40-12.json on k (40 -> 43) at equal n, distance tier, and weight class. Cell: weight-9plus x unrestricted.

Dead ends

  • The chain stopped after 4 accepted steps: the step-guard found no further deletion passing the free-bound/coverage/kill criteria.
  • Wave-1 lesson applied: the claim is priced by the theorem's terms, never by

    • the randomized search's output — at large n that search is far from tight (measured: weight-84 proposals where weight-28 witnesses exist).

Tools

GLM 5.3 Flash (matches provenance.model). Repo kit: GF(2) rank core, RIS surrogate, trusted verifier. The greedy-chain driver is fully specified above (~50 lines against the kit) and is deliberately not committed with this PR (one concern per PR).

Reproduction

1. Load codes/192-40-12.json from this PR's tree. 2. Delete the listed X-checks by 0-based index; keep every other check verbatim (indices refer to the parent's checks.X ordering). 3. k' = n - rank(H'_X) - rank(H_Z) = 43. 4. Verify: uv run python verify/qldpc_verify.py codes/192-43-12.json.

Parity checks

X-checks 76 (max weight 12) · Z-checks 80 (max weight 12)
H_X (76 checks, sparse supports)
[12, 21, 36, 49, 78, 88, 104, 124, 134, 147, 160, 191] [13, 22, 37, 50, 79, 89, 105, 125, 135, 148, 161, 176] [14, 23, 38, 51, 64, 90, 106, 126, 136, 149, 162, 177] [15, 24, 39, 52, 65, 91, 107, 127, 137, 150, 163, 178] [0, 25, 40, 53, 66, 92, 108, 112, 138, 151, 164, 179] [1, 26, 41, 54, 67, 93, 109, 113, 139, 152, 165, 180] [4, 29, 44, 57, 70, 80, 96, 116, 142, 155, 168, 183] [5, 30, 45, 58, 71, 81, 97, 117, 143, 156, 169, 184] [6, 31, 46, 59, 72, 82, 98, 118, 128, 157, 170, 185] [7, 16, 47, 60, 73, 83, 99, 119, 129, 158, 171, 186] [8, 17, 32, 61, 74, 84, 100, 120, 130, 159, 172, 187] [9, 18, 33, 62, 75, 85, 101, 121, 131, 144, 173, 188] [6, 26, 35, 50, 79, 92, 109, 118, 138, 148, 161, 190] [7, 27, 36, 51, 64, 93, 110, 119, 139, 149, 162, 191] [8, 28, 37, 52, 65, 94, 111, 120, 140, 150, 163, 176] [9, 29, 38, 53, 66, 95, 96, 121, 141, 151, 164, 177] [10, 30, 39, 54, 67, 80, 97, 122, 142, 152, 165, 178] [11, 31, 40, 55, 68, 81, 98, 123, 143, 153, 166, 179] [12, 16, 41, 56, 69, 82, 99, 124, 128, 154, 167, 180] [13, 17, 42, 57, 70, 83, 100, 125, 129, 155, 168, 181] [14, 18, 43, 58, 71, 84, 101, 126, 130, 156, 169, 182] [15, 19, 44, 59, 72, 85, 102, 127, 131, 157, 170, 183] [0, 20, 45, 60, 73, 86, 103, 112, 132, 158, 171, 184] [1, 21, 46, 61, 74, 87, 104, 113, 133, 159, 172, 185] [2, 22, 47, 62, 75, 88, 105, 114, 134, 144, 173, 186] [3, 23, 32, 63, 76, 89, 106, 115, 135, 145, 174, 187] [4, 24, 33, 48, 77, 90, 107, 116, 136, 146, 175, 188] [5, 25, 34, 49, 78, 91, 108, 117, 137, 147, 160, 189] [12, 22, 42, 51, 66, 95, 110, 125, 134, 154, 164, 177] [13, 23, 43, 52, 67, 80, 111, 126, 135, 155, 165, 178] [14, 24, 44, 53, 68, 81, 96, 127, 136, 156, 166, 179] [15, 25, 45, 54, 69, 82, 97, 112, 137, 157, 167, 180] [0, 26, 46, 55, 70, 83, 98, 113, 138, 158, 168, 181] [1, 27, 47, 56, 71, 84, 99, 114, 139, 159, 169, 182] [2, 28, 32, 57, 72, 85, 100, 115, 140, 144, 170, 183] [3, 29, 33, 58, 73, 86, 101, 116, 141, 145, 171, 184] [4, 30, 34, 59, 74, 87, 102, 117, 142, 146, 172, 185] [5, 31, 35, 60, 75, 88, 103, 118, 143, 147, 173, 186] [6, 16, 36, 61, 76, 89, 104, 119, 128, 148, 174, 187] [7, 17, 37, 62, 77, 90, 105, 120, 129, 149, 175, 188] [8, 18, 38, 63, 78, 91, 106, 121, 130, 150, 160, 189] [9, 19, 39, 48, 79, 92, 107, 122, 131, 151, 161, 190] [10, 20, 40, 49, 64, 93, 108, 123, 132, 152, 162, 191] [11, 21, 41, 50, 65, 94, 109, 124, 133, 153, 163, 176] [15, 28, 38, 58, 67, 82, 97, 126, 141, 150, 170, 180] [0, 29, 39, 59, 68, 83, 98, 127, 142, 151, 171, 181] [1, 30, 40, 60, 69, 84, 99, 112, 143, 152, 172, 182] [2, 31, 41, 61, 70, 85, 100, 113, 128, 153, 173, 183] [3, 16, 42, 62, 71, 86, 101, 114, 129, 154, 174, 184] [4, 17, 43, 63, 72, 87, 102, 115, 130, 155, 175, 185] [5, 18, 44, 48, 73, 88, 103, 116, 131, 156, 160, 186] [6, 19, 45, 49, 74, 89, 104, 117, 132, 157, 161, 187] [7, 20, 46, 50, 75, 90, 105, 118, 133, 158, 162, 188] [8, 21, 47, 51, 76, 91, 106, 119, 134, 159, 163, 189] [9, 22, 32, 52, 77, 92, 107, 120, 135, 144, 164, 190] [10, 23, 33, 53, 78, 93, 108, 121, 136, 145, 165, 191] [11, 24, 34, 54, 79, 94, 109, 122, 137, 146, 166, 176] [12, 25, 35, 55, 64, 95, 110, 123, 138, 147, 167, 177] [13, 26, 36, 56, 65, 80, 111, 124, 139, 148, 168, 178] [14, 27, 37, 57, 66, 81, 96, 125, 140, 149, 169, 179] [2, 31, 44, 54, 74, 83, 100, 113, 142, 157, 166, 186] [3, 16, 45, 55, 75, 84, 101, 114, 143, 158, 167, 187] [4, 17, 46, 56, 76, 85, 102, 115, 128, 159, 168, 188] [5, 18, 47, 57, 77, 86, 103, 116, 129, 144, 169, 189] [6, 19, 32, 58, 78, 87, 104, 117, 130, 145, 170, 190] [7, 20, 33, 59, 79, 88, 105, 118, 131, 146, 171, 191] [8, 21, 34, 60, 64, 89, 106, 119, 132, 147, 172, 176] [9, 22, 35, 61, 65, 90, 107, 120, 133, 148, 173, 177] [10, 23, 36, 62, 66, 91, 108, 121, 134, 149, 174, 178] [11, 24, 37, 63, 67, 92, 109, 122, 135, 150, 175, 179] [12, 25, 38, 48, 68, 93, 110, 123, 136, 151, 160, 180] [13, 26, 39, 49, 69, 94, 111, 124, 137, 152, 161, 181] [14, 27, 40, 50, 70, 95, 96, 125, 138, 153, 162, 182] [15, 28, 41, 51, 71, 80, 97, 126, 139, 154, 163, 183] [0, 29, 42, 52, 72, 81, 98, 127, 140, 155, 164, 184] [1, 30, 43, 53, 73, 82, 99, 112, 141, 156, 165, 185]
H_Z (80 checks, sparse supports)
[10, 19, 34, 63, 76, 86, 102, 122, 132, 145, 174, 189] [11, 20, 35, 48, 77, 87, 103, 123, 133, 146, 175, 190] [12, 21, 36, 49, 78, 88, 104, 124, 134, 147, 160, 191] [13, 22, 37, 50, 79, 89, 105, 125, 135, 148, 161, 176] [14, 23, 38, 51, 64, 90, 106, 126, 136, 149, 162, 177] [15, 24, 39, 52, 65, 91, 107, 127, 137, 150, 163, 178] [0, 25, 40, 53, 66, 92, 108, 112, 138, 151, 164, 179] [1, 26, 41, 54, 67, 93, 109, 113, 139, 152, 165, 180] [2, 27, 42, 55, 68, 94, 110, 114, 140, 153, 166, 181] [3, 28, 43, 56, 69, 95, 111, 115, 141, 154, 167, 182] [4, 29, 44, 57, 70, 80, 96, 116, 142, 155, 168, 183] [5, 30, 45, 58, 71, 81, 97, 117, 143, 156, 169, 184] [6, 31, 46, 59, 72, 82, 98, 118, 128, 157, 170, 185] [7, 16, 47, 60, 73, 83, 99, 119, 129, 158, 171, 186] [8, 17, 32, 61, 74, 84, 100, 120, 130, 159, 172, 187] [9, 18, 33, 62, 75, 85, 101, 121, 131, 144, 173, 188] [6, 26, 35, 50, 79, 92, 109, 118, 138, 148, 161, 190] [7, 27, 36, 51, 64, 93, 110, 119, 139, 149, 162, 191] [8, 28, 37, 52, 65, 94, 111, 120, 140, 150, 163, 176] [9, 29, 38, 53, 66, 95, 96, 121, 141, 151, 164, 177] [10, 30, 39, 54, 67, 80, 97, 122, 142, 152, 165, 178] [11, 31, 40, 55, 68, 81, 98, 123, 143, 153, 166, 179] [12, 16, 41, 56, 69, 82, 99, 124, 128, 154, 167, 180] [13, 17, 42, 57, 70, 83, 100, 125, 129, 155, 168, 181] [14, 18, 43, 58, 71, 84, 101, 126, 130, 156, 169, 182] [15, 19, 44, 59, 72, 85, 102, 127, 131, 157, 170, 183] [0, 20, 45, 60, 73, 86, 103, 112, 132, 158, 171, 184] [1, 21, 46, 61, 74, 87, 104, 113, 133, 159, 172, 185] [2, 22, 47, 62, 75, 88, 105, 114, 134, 144, 173, 186] [3, 23, 32, 63, 76, 89, 106, 115, 135, 145, 174, 187] [4, 24, 33, 48, 77, 90, 107, 116, 136, 146, 175, 188] [5, 25, 34, 49, 78, 91, 108, 117, 137, 147, 160, 189] [12, 22, 42, 51, 66, 95, 110, 125, 134, 154, 164, 177] [13, 23, 43, 52, 67, 80, 111, 126, 135, 155, 165, 178] [14, 24, 44, 53, 68, 81, 96, 127, 136, 156, 166, 179] [15, 25, 45, 54, 69, 82, 97, 112, 137, 157, 167, 180] [0, 26, 46, 55, 70, 83, 98, 113, 138, 158, 168, 181] [1, 27, 47, 56, 71, 84, 99, 114, 139, 159, 169, 182] [2, 28, 32, 57, 72, 85, 100, 115, 140, 144, 170, 183] [3, 29, 33, 58, 73, 86, 101, 116, 141, 145, 171, 184] [4, 30, 34, 59, 74, 87, 102, 117, 142, 146, 172, 185] [5, 31, 35, 60, 75, 88, 103, 118, 143, 147, 173, 186] [6, 16, 36, 61, 76, 89, 104, 119, 128, 148, 174, 187] [7, 17, 37, 62, 77, 90, 105, 120, 129, 149, 175, 188] [8, 18, 38, 63, 78, 91, 106, 121, 130, 150, 160, 189] [9, 19, 39, 48, 79, 92, 107, 122, 131, 151, 161, 190] [10, 20, 40, 49, 64, 93, 108, 123, 132, 152, 162, 191] [11, 21, 41, 50, 65, 94, 109, 124, 133, 153, 163, 176] [15, 28, 38, 58, 67, 82, 97, 126, 141, 150, 170, 180] [0, 29, 39, 59, 68, 83, 98, 127, 142, 151, 171, 181] [1, 30, 40, 60, 69, 84, 99, 112, 143, 152, 172, 182] [2, 31, 41, 61, 70, 85, 100, 113, 128, 153, 173, 183] [3, 16, 42, 62, 71, 86, 101, 114, 129, 154, 174, 184] [4, 17, 43, 63, 72, 87, 102, 115, 130, 155, 175, 185] [5, 18, 44, 48, 73, 88, 103, 116, 131, 156, 160, 186] [6, 19, 45, 49, 74, 89, 104, 117, 132, 157, 161, 187] [7, 20, 46, 50, 75, 90, 105, 118, 133, 158, 162, 188] [8, 21, 47, 51, 76, 91, 106, 119, 134, 159, 163, 189] [9, 22, 32, 52, 77, 92, 107, 120, 135, 144, 164, 190] [10, 23, 33, 53, 78, 93, 108, 121, 136, 145, 165, 191] [11, 24, 34, 54, 79, 94, 109, 122, 137, 146, 166, 176] [12, 25, 35, 55, 64, 95, 110, 123, 138, 147, 167, 177] [13, 26, 36, 56, 65, 80, 111, 124, 139, 148, 168, 178] [14, 27, 37, 57, 66, 81, 96, 125, 140, 149, 169, 179] [2, 31, 44, 54, 74, 83, 100, 113, 142, 157, 166, 186] [3, 16, 45, 55, 75, 84, 101, 114, 143, 158, 167, 187] [4, 17, 46, 56, 76, 85, 102, 115, 128, 159, 168, 188] [5, 18, 47, 57, 77, 86, 103, 116, 129, 144, 169, 189] [6, 19, 32, 58, 78, 87, 104, 117, 130, 145, 170, 190] [7, 20, 33, 59, 79, 88, 105, 118, 131, 146, 171, 191] [8, 21, 34, 60, 64, 89, 106, 119, 132, 147, 172, 176] [9, 22, 35, 61, 65, 90, 107, 120, 133, 148, 173, 177] [10, 23, 36, 62, 66, 91, 108, 121, 134, 149, 174, 178] [11, 24, 37, 63, 67, 92, 109, 122, 135, 150, 175, 179] [12, 25, 38, 48, 68, 93, 110, 123, 136, 151, 160, 180] [13, 26, 39, 49, 69, 94, 111, 124, 137, 152, 161, 181] [14, 27, 40, 50, 70, 95, 96, 125, 138, 153, 162, 182] [15, 28, 41, 51, 71, 80, 97, 126, 139, 154, 163, 183] [0, 29, 42, 52, 72, 81, 98, 127, 140, 155, 164, 184] [1, 30, 43, 53, 73, 82, 99, 112, 141, 156, 165, 185]
Code ID 192-43-12 · download JSON · raw on GitHub