← back to the board
[[92,2,14]] d ≤
n
92
k
2
d
14
kd²/n
4.261
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[1, 4, 8, 9, 41, 43, 47, 72, 74, 76, 80, 85, 86, 89]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[1, 8, 9, 16, 28, 49, 50, 54, 55, 64, 74, 75, 76, 82]
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 8 · H_Z 8
qubit degrees H_X 3–4 (mean 3.913) · H_Z 3–4 (mean 3.913)
trapping sets H_X (1,3)×8 (2,3)×20 (3,3)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 8 (1,4): 84 (2,3): 20 (2,4): 210 (2,5): 108 (2,6): 702 (3,3): 32 (3,4): 336 (3,5): 726 (3,6): 4750 (3,7): 1631 (3,8): 7504 (3,9): 135 (3,10): 540
trapping sets H_Z (1,3)×8 (2,3)×20 (3,3)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 8 (1,4): 84 (2,3): 20 (2,4): 210 (2,5): 108 (2,6): 702 (3,3): 32 (3,4): 336 (3,5): 726 (3,6): 4750 (3,7): 1631 (3,8): 7504 (3,9): 135 (3,10): 540

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 67c1247a7ca45da389d67f33 (https://qecdb.org/codes/67c1247a7ca45da389d67f33); the record's H field is read as binary symplectic rows and split by rowspace (rank(A|B) = rank(A) + rank(B)), keeping the lightest independent basis of each part; n, k and the check weight recomputed here, witnesses from the kit's random search at the trials below. Database description: 2BGA; database distance claim d=14 (lower 14, upper 14).
model Mimo-V2.6-Free (claimed, not verified)
date 2026-09-27
notes Equivalence assessed against the board: no exact stabilizer-group fingerprint match and no Weisfeiler-Leman signature match to any existing entry.
family generalized bicycle (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

[[92,2,14]] — qecdb.org 2BGA code, reconstructed from its H field

Direction & hypothesis

Track cell weight-8 × unrestricted, CSS board. The hypothesis was that the public parameter tables sit in front of this board: qecdb.org indexes 15,181 CSS codes with matrices, and sweeping all of them against the board's own 1,513 CSS entries should surface records whose parameters dominate an entry -- if the matrices survive reconstruction and the distance readings hold up. This candidate is record 67c1247a7ca45da389d67f33.

What was searched

  • Mirrored the whole CSS half of qecdb.org (/api/codes?cssOnly=1, 200 per

    • page): 15,181 records, n ≤ 300, d ≤ 17. A parameter pre-filter against the board left 1,527 records (dropped 136 over the verifier's weight-32 cap and 13,654 dominated on (n, k, d, w) as listed); every one of those was then fetched in full for its stabilizer generators.

  • Reconstruction: the record's H field is binary symplectic rows; split it

    • by rowspace (rank(A|B) = rank(A) + rank(B)) and keep the *lightest* independent basis of each side. Two traps this avoids -- a plain RREF inflates max check weight, and self-dual records store mixed (Y-bearing) generators of an otherwise CSS code. n, k and w were recomputed here; all 1,527 agreed with the database on k and on check weight.

  • Screening: every survivor at 50,000 RIS trials/side, bit-packed backend,

    • seed derived from the code's own fingerprint so a reading is reproducible. 608 records still beat a board entry; 314 of the 1,510 reconstructed survivors turn out to be direct sums, so connectivity is checked *before* anything is picked (104 of 221 remain).

  • Selection: cell-restricted advance (no dominator, at least one in-cell entry

    • beaten), ranked tracks (w ≤ 8), one code, a gain on something other than d, and no entry beaten on d alone (the d-only win is the inflation pattern). One record per (n, k, w) tuple.

  • This candidate: [[92,2,14]] at w = 8, record 67c1247a7ca45da389d67f33; qecdb description

    • "2BGA", database distance claim d = 14 (lower 14, upper 14).

Evidence trail

  • Ladder: 14 at deep2 → 14 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [1, 12, 15, 16, 23, 29, 30, 33, 36, 44, 51, 66, 82, 85] and Z witness [0, 3, 5, 20, 26, 30, 33, 34, 46, 50, 63, 68, 69, 84], both of weight 14.

  • Claim, precisely: a witness-backed upper bound d ≤ 14. Nothing here

    • certifies the exact distance; verify/validate_candidate.py also ran its own refutation search on the packaged document (seed recorded in the verdict beside this candidate) and found no lighter logical.

  • Board comparison (cell-restricted): beats 1 entry on n: 96-2-14.json. Efficiency k d² / n = 4.26.
  • Verdict: passed: true, not refuted, not a duplicate of a board entry;

    • labels: advances the weight-8 x unrestricted board; literature novelty UNVERIFIED.

  • The full validator verdict (refutation seed, fingerprint, cell, labels) is

    • kept alongside the submission in this run's staging area for the reviewer.

Dead ends

  • Direct sums: 314 of 1,510 reconstructed records are not one code (the

    • verifier's tanner_connected / stabilizer_group_connected reject them). Ten of the first 35 picks fell here; their 947 blocks were re-screened as separate candidates.

  • d-only wins: candidates whose only strict axis over some entry was d were

    • rejected as distance inflation (three already-packaged ones were unstaged for this: [[48,18,4]], [[56,6,8]], [[84,44,3]]).

  • A second [[144,12,12]] w = 6 record passed the gate but ties the board's own

    • entry on all four axes, so it was unstaged: same parameters, no frontier gain.

  • ECZ (errorcorrectionzoo.org/all, 1,185 entries) is an index of parameters

    • and references only -- no matrices, so it cannot feed this pipeline; it is still the right place to check literature novelty later.

Tools

opencode/mimo-v2.6-flash-free (opencode) driving the repo's own kit (research/kit/: css, surrogate, submit, search) and the untouched verify/ stack; the bit-packed C++ RIS backend behind distance_rand(backend="fast") for the screens, numpy for the packaging witnesses. Whole scrape: ~25 minutes wall clock on a 10-core laptop.

Reproduction

1. GET https://qecdb.org/api/codes?cssOnly=1&pageSize=200&page=N for the list, GET https://qecdb.org/codes/67c1247a7ca45da389d67f33 (the API path /api/codes/67c1247a7ca45da389d67f33) for the generators. 2. Parse H into binary symplectic rows; set A = rows with no Y, B = rows with Y only in their X half ... in practice: build the [A|B] matrix and split by rowspace, taking the lightest independent basis of each part (rank over GF(2), greedy lightest-first). 3. Verify H_X H_Zᵀ = 0, recompute k = n − rank(H_X) − rank(H_Z), take w = max row weight. 4. Screen: `distance_rand(HX, HZ, trials=50000, seed=int(fingerprint,16) % 2**31, backend="fast")`, then the same at 2,000,000 trials, then package with submit.make_submission(HX, HZ, trials=8000, seed=0). 5. Run verify/validate_candidate.py on the packaged document; keep only passed: true.

Parity checks

X-checks 45 (max weight 8) · Z-checks 45 (max weight 8)
H_X (45 checks, sparse supports)
[0, 25, 26, 38, 46, 52, 84, 90] [1, 2, 3, 15, 47, 61, 68, 75] [2, 20, 32, 33, 48, 53, 60, 66] [3, 21, 33, 35, 49, 54, 61, 67] [4, 22, 35, 36, 50, 55, 62, 68] [5, 24, 36, 37, 51, 56, 63, 70] [6, 25, 37, 38, 52, 57, 58, 71] [7, 26, 38, 39, 46, 53, 59, 72] [8, 27, 39, 40, 54, 60, 73, 80] [9, 18, 28, 40, 55, 61, 74, 87] [1, 10, 18, 29, 56, 62, 75, 88] [1, 11, 23, 30, 57, 63, 76, 89] [12, 24, 25, 37, 51, 58, 83, 89] [13, 19, 23, 31, 58, 59, 77, 90] [14, 19, 20, 32, 46, 60, 78, 91] [15, 20, 21, 33, 48, 61, 79, 80] [16, 21, 22, 35, 49, 62, 81, 87] [17, 22, 24, 36, 50, 63, 82, 88] [2, 14, 18, 45, 60, 64, 67, 74] [4, 5, 17, 19, 63, 65, 71, 77] [5, 6, 12, 20, 58, 66, 72, 78] [0, 6, 7, 21, 46, 67, 73, 79] [7, 8, 22, 34, 68, 74, 80, 81] [3, 4, 16, 23, 62, 69, 70, 76] [8, 9, 24, 41, 70, 75, 82, 87] [9, 10, 25, 42, 71, 76, 83, 88] [10, 11, 26, 43, 72, 77, 84, 89] [11, 13, 27, 44, 73, 78, 85, 90] [13, 14, 28, 45, 74, 79, 86, 91] [2, 14, 15, 29, 48, 64, 75, 81] [3, 15, 16, 30, 47, 49, 76, 82] [4, 16, 17, 31, 50, 69, 77, 83] [5, 12, 17, 32, 51, 65, 78, 84] [0, 6, 12, 33, 52, 66, 79, 85] [26, 27, 34, 39, 53, 80, 85, 91] [0, 7, 34, 35, 53, 67, 81, 86] [8, 34, 36, 41, 54, 64, 68, 82] [9, 37, 41, 42, 47, 55, 70, 83] [10, 38, 42, 43, 56, 69, 71, 84] [11, 39, 43, 44, 57, 65, 72, 85] [13, 40, 44, 45, 59, 66, 73, 86] [27, 28, 40, 41, 48, 54, 86, 87] [18, 28, 29, 42, 49, 55, 64, 88] [1, 29, 30, 43, 47, 50, 56, 89] [23, 30, 31, 44, 51, 57, 69, 90]
H_Z (45 checks, sparse supports)
[0, 7, 14, 21, 46, 67, 79, 81] [1, 30, 37, 43, 47, 56, 57, 89] [2, 15, 29, 41, 47, 48, 64, 75] [3, 16, 30, 42, 47, 49, 69, 76] [4, 17, 31, 43, 50, 65, 69, 77] [5, 12, 32, 44, 51, 65, 66, 78] [0, 6, 33, 45, 52, 66, 67, 79] [2, 7, 34, 35, 53, 67, 68, 81] [3, 8, 36, 41, 54, 68, 70, 82] [4, 9, 37, 42, 55, 70, 71, 83] [5, 10, 38, 43, 56, 71, 72, 84] [6, 11, 39, 44, 57, 72, 73, 85] [6, 12, 13, 20, 58, 66, 78, 79] [7, 13, 40, 45, 59, 73, 74, 86] [2, 8, 14, 18, 60, 64, 74, 75] [1, 3, 9, 15, 47, 61, 75, 76] [4, 10, 16, 23, 62, 69, 76, 77] [5, 11, 17, 19, 63, 65, 77, 78] [18, 29, 36, 42, 55, 56, 64, 88] [19, 32, 39, 45, 59, 60, 65, 91] [2, 20, 33, 40, 48, 60, 61, 66] [3, 18, 21, 35, 49, 61, 62, 67] [1, 4, 22, 36, 50, 62, 63, 68] [23, 31, 38, 44, 57, 59, 69, 90] [5, 23, 24, 37, 51, 58, 63, 70] [6, 19, 25, 38, 46, 52, 58, 71] [7, 20, 26, 39, 46, 53, 72, 80] [8, 21, 27, 40, 54, 73, 80, 87] [9, 18, 22, 28, 55, 74, 87, 88] [1, 10, 24, 29, 56, 75, 88, 89] [11, 23, 25, 30, 57, 76, 89, 90] [13, 19, 26, 31, 59, 77, 90, 91] [14, 20, 27, 32, 48, 60, 78, 91] [15, 21, 28, 33, 48, 49, 61, 79] [8, 15, 22, 34, 68, 80, 81, 82] [16, 22, 29, 35, 49, 50, 62, 81] [17, 24, 30, 36, 50, 51, 63, 82] [12, 25, 31, 37, 51, 52, 58, 83] [0, 26, 32, 38, 46, 52, 53, 84] [27, 33, 34, 39, 53, 54, 80, 85] [28, 35, 40, 41, 54, 55, 86, 87] [9, 16, 24, 41, 70, 82, 83, 87] [10, 17, 25, 42, 71, 83, 84, 88] [11, 12, 26, 43, 72, 84, 85, 89] [0, 13, 27, 44, 73, 85, 86, 90]
Code ID 92-2-14 · download JSON · raw on GitHub