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[[102,22,9]] d ≤
n
102
k
22
d
9
kd²/n
17.471
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)
[3, 28, 32, 48, 51, 65, 84, 88, 97]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[16, 21, 38, 48, 54, 61, 68, 77, 90]
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 3.137) · H_Z 2–4 (mean 3.137)
trapping sets H_X (1,2)×18 (2,2)×21 (3,2)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 18 (1,3): 52 (1,4): 32 (2,2): 21 (2,3): 119 (2,4): 358 (2,5): 439 (2,6): 183 (3,2): 40 (3,3): 257 (3,4): 1448 (3,5): 3815 (3,6): 5836 (3,7): 4705 (3,8): 1933 (3,9): 551 (3,10): 149
trapping sets H_Z (1,2)×18 (2,2)×21 (3,2)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 18 (1,3): 52 (1,4): 32 (2,2): 21 (2,3): 119 (2,4): 358 (2,5): 439 (2,6): 183 (3,2): 40 (3,3): 257 (3,4): 1448 (3,5): 3815 (3,6): 5836 (3,7): 4705 (3,8): 1933 (3,9): 551 (3,10): 149

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 6a422a11d15d481af09df8a3 (https://qecdb.org/codes/6a422a11d15d481af09df8a3); 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

[[102,22,9]] — 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 6a422a11d15d481af09df8a3.

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: [[102,22,9]] at w = 8, record 6a422a11d15d481af09df8a3; 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 deep → 9 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [35, 41, 47, 54, 56, 67, 73, 80, 90] and Z witness [10, 18, 37, 38, 58, 73, 84, 93, 98], 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 2 entries on d, k, n: 128-21-8.json, 144-18-9.json. Efficiency k d² / n = 17.47.
  • 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/6a422a11d15d481af09df8a3 (the API path /api/codes/6a422a11d15d481af09df8a3) 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 40 (max weight 8) · Z-checks 40 (max weight 8)
H_X (40 checks, sparse supports)
[14, 24, 39, 41, 51, 59, 79, 85] [16, 20, 35, 43, 52, 61, 81, 87] [17, 27, 42, 44, 53, 62, 82, 88] [18, 22, 37, 45, 54, 63, 77, 83] [19, 23, 38, 46, 55, 64, 84, 90] [20, 30, 45, 47, 56, 65, 85, 91] [21, 25, 40, 48, 57, 66, 80, 86] [22, 26, 41, 49, 58, 67, 87, 93] [0, 23, 33, 48, 59, 68, 88, 94] [24, 28, 43, 50, 60, 69, 83, 89] [1, 25, 29, 44, 61, 70, 90, 96] [2, 26, 36, 50, 62, 71, 91, 97] [3, 27, 31, 46, 63, 72, 86, 92] [4, 28, 32, 47, 64, 73, 93, 99] [3, 5, 29, 39, 65, 74, 94, 100] [6, 30, 34, 49, 66, 75, 89, 95] [0, 7, 31, 35, 67, 76, 96, 101] [6, 8, 32, 42, 52, 68, 77, 97] [1, 9, 33, 37, 69, 78, 92, 98] [2, 10, 34, 38, 54, 70, 79, 99] [9, 11, 35, 45, 55, 71, 80, 100] [4, 12, 36, 40, 51, 72, 81, 95] [5, 13, 37, 41, 57, 73, 82, 101] [12, 14, 38, 48, 52, 58, 74, 83] [7, 15, 39, 43, 53, 75, 84, 98] [8, 16, 40, 44, 54, 60, 76, 85] [15, 17, 41, 50, 55, 61, 77, 86] [10, 18, 42, 46, 51, 56, 78, 87] [11, 19, 43, 47, 57, 63, 79, 88] [3, 18, 20, 44, 58, 64, 80, 89] [13, 21, 45, 49, 53, 59, 81, 90] [0, 14, 22, 46, 60, 66, 82, 91] [6, 21, 23, 47, 61, 67, 83, 92] [1, 16, 24, 48, 56, 62, 84, 93] [2, 17, 25, 49, 63, 69, 85, 94] [0, 9, 24, 26, 64, 70, 86, 95] [4, 19, 27, 50, 59, 65, 87, 96] [2, 12, 27, 29, 67, 73, 89, 98] [5, 15, 30, 32, 51, 70, 76, 92] [8, 18, 33, 35, 53, 73, 79, 95]
H_Z (40 checks, sparse supports)
[0, 21, 27, 41, 59, 67, 82, 86] [1, 17, 23, 43, 61, 69, 84, 88] [2, 24, 30, 44, 62, 70, 85, 89] [3, 19, 25, 45, 63, 65, 80, 90] [4, 20, 26, 46, 64, 72, 87, 91] [5, 27, 33, 47, 65, 73, 88, 92] [6, 22, 28, 48, 66, 68, 83, 93] [7, 23, 29, 49, 67, 75, 90, 94] [0, 8, 30, 36, 68, 76, 91, 95] [9, 25, 31, 50, 69, 71, 86, 96] [1, 10, 26, 32, 70, 78, 93, 97] [2, 11, 33, 39, 71, 79, 94, 98] [3, 12, 28, 34, 72, 74, 89, 99] [4, 13, 29, 35, 73, 81, 96, 100] [5, 14, 36, 42, 51, 74, 82, 97] [6, 15, 31, 37, 75, 77, 92, 101] [7, 16, 32, 38, 52, 76, 84, 99] [8, 17, 39, 45, 53, 77, 85, 100] [9, 18, 34, 40, 54, 78, 80, 95] [10, 19, 35, 41, 55, 79, 87, 101] [11, 20, 42, 48, 52, 56, 80, 88] [12, 21, 37, 43, 57, 81, 83, 98] [13, 22, 38, 44, 54, 58, 82, 90] [14, 23, 45, 50, 55, 59, 83, 91] [15, 24, 40, 46, 51, 60, 84, 86] [16, 25, 41, 47, 57, 61, 85, 93] [3, 17, 26, 48, 58, 62, 86, 94] [18, 27, 43, 49, 53, 63, 87, 89] [0, 19, 28, 44, 60, 64, 88, 96] [6, 20, 29, 50, 61, 65, 89, 97] [1, 21, 30, 46, 56, 66, 90, 92] [2, 22, 31, 47, 63, 67, 91, 99] [3, 9, 23, 32, 64, 68, 92, 100] [4, 24, 33, 49, 59, 69, 93, 95] [0, 5, 25, 34, 66, 70, 94, 101] [6, 12, 26, 35, 52, 67, 71, 95] [1, 7, 27, 36, 62, 72, 96, 98] [9, 15, 29, 38, 55, 70, 74, 98] [12, 18, 32, 41, 51, 58, 73, 77] [15, 21, 35, 44, 53, 61, 76, 80]
Code ID 102-22-9 · download JSON · raw on GitHub