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[[102,18,9]] d ≤
n
102
k
18
d
9
kd²/n
14.294
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[0, 2, 21, 30, 37, 41, 42, 71, 86]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[5, 10, 23, 33, 38, 43, 59, 64, 80]
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 6 · H_Z 6 (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 2–4 (mean 3.294) · H_Z 2–4 (mean 3.294)
trapping sets H_X (1,2)×14 (2,2)×9 (3,1)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 14 (1,3): 44 (1,4): 44 (2,2): 9 (2,3): 72 (2,4): 272 (2,5): 520 (2,6): 303 (3,1): 3 (3,2): 12 (3,3): 154 (3,4): 820 (3,5): 2922 (3,6): 5753 (3,7): 6435 (3,8): 3195 (3,9): 830 (3,10): 274
trapping sets H_Z (1,2)×14 (2,2)×9 (3,1)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 14 (1,3): 44 (1,4): 44 (2,2): 9 (2,3): 70 (2,4): 278 (2,5): 514 (2,6): 305 (3,1): 3 (3,2): 14 (3,3): 140 (3,4): 852 (3,5): 2892 (3,6): 5766 (3,7): 6429 (3,8): 3209 (3,9): 812 (3,10): 281

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 6a41415955928fd7347649b1 (https://qecdb.org/codes/6a41415955928fd7347649b1); 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 w=8 wa=4 wb=4 G=SmallGroup(51, 1); arXiv:2306.16400; database distance claim d=9 (lower None, upper None).
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

[[102,18,9]] — qecdb.org 2BGA w=8 wa=4 wb=4 G=SmallGroup(51, 1); arXiv:2306.16400 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 6a41415955928fd7347649b1.

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: [[102,18,9]] at w = 8, record 6a41415955928fd7347649b1; qecdb description

    • "2BGA w=8 wa=4 wb=4 G=SmallGroup(51, 1); arXiv:2306.16400", database distance claim d = ? (lower ?, upper ?).

Evidence trail

  • Ladder: 9 at deep2 → 9 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [29, 45, 51, 60, 73, 80, 95, 96, 98] and Z witness [36, 50, 51, 69, 72, 80, 81, 85, 92], both of weight 9.

  • Claim, precisely: a witness-backed upper bound d ≤ 9. 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: 144-18-9.json. Efficiency k d² / n = 14.29.
  • 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/6a41415955928fd7347649b1 (the API path /api/codes/6a41415955928fd7347649b1) 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 42 (max weight 8) · Z-checks 42 (max weight 8)
H_X (42 checks, sparse supports)
[9, 24, 27, 37, 55, 60, 90, 99] [5, 20, 23, 39, 56, 57, 86, 95] [12, 27, 30, 40, 58, 63, 93, 101] [7, 22, 25, 35, 53, 58, 88, 97] [8, 23, 26, 42, 59, 60, 89, 98] [15, 30, 33, 43, 54, 61, 66, 96] [10, 25, 28, 38, 56, 61, 91, 100] [11, 26, 29, 45, 51, 62, 63, 92] [18, 33, 36, 46, 57, 64, 69, 99] [13, 28, 31, 41, 52, 59, 64, 94] [14, 29, 32, 48, 53, 65, 66, 95] [21, 36, 39, 49, 60, 67, 72, 101] [16, 31, 34, 44, 55, 62, 67, 97] [17, 32, 35, 50, 56, 68, 69, 98] [1, 24, 39, 42, 54, 63, 70, 75] [19, 34, 37, 47, 58, 65, 70, 100] [3, 20, 35, 38, 51, 59, 71, 72] [4, 27, 42, 45, 57, 66, 73, 78] [0, 22, 37, 40, 52, 61, 68, 73] [6, 23, 38, 41, 53, 62, 74, 75] [7, 30, 45, 48, 60, 69, 76, 81] [2, 25, 40, 43, 55, 64, 71, 76] [9, 26, 41, 44, 56, 65, 77, 78] [10, 33, 48, 50, 63, 72, 79, 84] [5, 28, 43, 46, 58, 67, 74, 79] [12, 29, 44, 47, 59, 68, 80, 81] [3, 13, 36, 50, 66, 75, 82, 87] [8, 31, 46, 49, 61, 70, 77, 82] [0, 15, 32, 47, 62, 71, 83, 84] [3, 6, 16, 39, 69, 78, 85, 90] [1, 11, 34, 49, 64, 73, 80, 85] [0, 2, 18, 35, 65, 74, 86, 87] [6, 9, 19, 42, 72, 81, 88, 93] [1, 4, 14, 37, 67, 76, 83, 88] [2, 5, 21, 38, 68, 77, 89, 90] [9, 12, 22, 45, 75, 84, 91, 96] [4, 7, 17, 40, 70, 79, 86, 91] [5, 8, 24, 41, 71, 80, 92, 93] [12, 15, 25, 48, 78, 87, 94, 99] [7, 10, 20, 43, 73, 82, 89, 94] [15, 18, 28, 50, 81, 90, 97, 101] [3, 18, 21, 31, 54, 84, 93, 100]
H_Z (42 checks, sparse supports)
[7, 16, 46, 50, 69, 79, 82, 97] [9, 18, 47, 48, 65, 81, 84, 99] [3, 10, 19, 49, 72, 82, 85, 100] [5, 14, 44, 49, 67, 77, 80, 95] [0, 12, 21, 50, 68, 84, 87, 101] [1, 6, 13, 22, 52, 75, 85, 88] [1, 8, 17, 47, 70, 80, 83, 98] [2, 3, 15, 24, 54, 71, 87, 90] [4, 9, 16, 25, 55, 78, 88, 91] [0, 4, 11, 20, 51, 73, 83, 86] [5, 6, 18, 27, 57, 74, 90, 93] [7, 12, 19, 28, 58, 81, 91, 94] [2, 7, 14, 23, 53, 76, 86, 89] [8, 9, 21, 30, 60, 77, 93, 96] [10, 15, 22, 31, 61, 84, 94, 97] [5, 10, 17, 26, 56, 79, 89, 92] [11, 12, 24, 33, 63, 80, 96, 99] [13, 18, 25, 34, 64, 87, 97, 100] [8, 13, 20, 29, 59, 82, 92, 95] [14, 15, 27, 36, 66, 83, 99, 101] [16, 21, 28, 37, 52, 67, 90, 100] [11, 16, 23, 32, 62, 85, 95, 98] [17, 18, 30, 39, 54, 69, 86, 101] [19, 24, 31, 40, 52, 55, 70, 93] [14, 19, 26, 35, 51, 65, 88, 98] [20, 21, 33, 42, 54, 57, 72, 89] [22, 27, 34, 43, 55, 58, 73, 96] [17, 22, 29, 38, 51, 53, 68, 91] [23, 24, 36, 45, 57, 60, 75, 92] [25, 30, 37, 46, 58, 61, 76, 99] [20, 25, 32, 41, 53, 56, 71, 94] [26, 27, 39, 48, 60, 63, 78, 95] [28, 33, 40, 49, 61, 64, 79, 101] [23, 28, 35, 44, 56, 59, 74, 97] [29, 30, 42, 50, 63, 66, 81, 98] [1, 31, 36, 43, 54, 64, 67, 82] [26, 31, 38, 47, 59, 62, 77, 100] [3, 32, 33, 45, 51, 66, 69, 84] [4, 34, 39, 46, 57, 67, 70, 85] [0, 29, 34, 41, 52, 62, 65, 80] [7, 37, 42, 49, 60, 70, 73, 88] [1, 10, 40, 45, 63, 73, 76, 91]
Code ID 102-18-9 · download JSON · raw on GitHub