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[[128,22,10]] d ≤
n
128
k
22
d
10
kd²/n
17.188
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[28, 29, 41, 47, 52, 53, 64, 114, 117, 122]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[6, 10, 33, 47, 75, 76, 86, 95, 104, 117]
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 1–4 (mean 3.312) · H_Z 1–4 (mean 3.312)
trapping sets H_X (1,1)×2 (2,1)×1 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 2 (1,2): 19 (1,3): 44 (1,4): 63 (2,1): 1 (2,2): 11 (2,3): 100 (2,4): 287 (2,5): 575 (2,6): 510 (3,1): 1 (3,2): 23 (3,3): 190 (3,4): 896 (3,5): 3142 (3,6): 6469 (3,7): 8634 (3,8): 5853 (3,9): 1041 (3,10): 574
trapping sets H_Z (1,1)×2 (2,2)×9 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 2 (1,2): 12 (1,3): 58 (1,4): 56 (2,2): 9 (2,3): 83 (2,4): 325 (2,5): 661 (2,6): 406 (3,1): 1 (3,2): 15 (3,3): 157 (3,4): 924 (3,5): 3233 (3,6): 7333 (3,7): 8894 (3,8): 4550 (3,9): 1033 (3,10): 420

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 6a41415955928fd7347649b9 (https://qecdb.org/codes/6a41415955928fd7347649b9); 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(64, 26); arXiv:2306.16400; database distance claim d=10 (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

[[128,22,10]] — qecdb.org 2BGA w=8 wa=4 wb=4 G=SmallGroup(64, 26); 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 6a41415955928fd7347649b9.

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: [[128,22,10]] at w = 8, record 6a41415955928fd7347649b9; qecdb description

    • "2BGA w=8 wa=4 wb=4 G=SmallGroup(64, 26); arXiv:2306.16400", database distance claim d = ? (lower ?, upper ?).

Evidence trail

  • Ladder: 10 at deep → 10 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [28, 29, 41, 47, 52, 53, 64, 114, 117, 122] and Z witness [6, 50, 53, 57, 69, 74, 77, 105, 122, 127], both of weight 10.

  • Claim, precisely: a witness-backed upper bound d ≤ 10. 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 10 entries on d, k, n: 128-13-10.json, 128-21-8.json, 140-10-10.json, 144-16-10.json, 144-18-9.json, 161-12-10.json. Efficiency k d² / n = 17.19.
  • 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/6a41415955928fd7347649b9 (the API path /api/codes/6a41415955928fd7347649b9) 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 53 (max weight 8) · Z-checks 53 (max weight 8)
H_X (53 checks, sparse supports)
[17, 20, 27, 46, 78, 85, 94, 124] [6, 27, 30, 53, 88, 95, 103, 126] [11, 33, 36, 43, 83, 95, 101, 110] [6, 11, 39, 58, 97, 104, 113, 127] [6, 25, 38, 48, 75, 99, 105, 111] [18, 28, 41, 60, 64, 79, 87, 115] [4, 23, 40, 50, 69, 73, 101, 109] [15, 18, 43, 46, 93, 104, 111, 117] [11, 18, 49, 55, 77, 105, 107, 114] [11, 20, 34, 48, 82, 109, 115, 118] [21, 28, 51, 62, 65, 89, 96, 120] [0, 9, 32, 50, 74, 80, 111, 116] [15, 25, 28, 53, 102, 114, 118, 122] [15, 30, 52, 57, 69, 89, 115, 119] [31, 34, 44, 55, 65, 66, 84, 124] [1, 13, 54, 59, 74, 78, 87, 105] [18, 20, 30, 44, 92, 116, 120, 123] [21, 31, 56, 63, 67, 98, 106, 125] [0, 1, 16, 42, 81, 90, 118, 121] [21, 39, 47, 49, 68, 71, 95, 100] [0, 7, 56, 61, 65, 83, 88, 119] [3, 8, 19, 45, 66, 70, 93, 110] [21, 25, 34, 58, 64, 112, 119, 123] [25, 36, 39, 57, 74, 98, 120, 124] [37, 40, 44, 60, 67, 71, 94, 126] [2, 3, 23, 59, 81, 88, 96, 115] [28, 30, 37, 39, 66, 85, 121, 125] [13, 31, 40, 61, 68, 72, 99, 108] [1, 3, 26, 35, 83, 91, 100, 123] [31, 41, 49, 54, 73, 76, 104, 110] [1, 4, 12, 61, 67, 93, 97, 124] [5, 8, 29, 52, 71, 75, 102, 117] [34, 36, 46, 49, 81, 108, 125, 126] [37, 47, 50, 62, 72, 76, 103, 127] [2, 7, 8, 32, 91, 97, 106, 120] [37, 51, 53, 58, 70, 73, 77, 111] [2, 9, 62, 63, 71, 85, 93, 99] [10, 12, 22, 35, 68, 75, 79, 109] [40, 41, 51, 59, 80, 86, 114, 117] [3, 4, 9, 22, 72, 102, 107, 126] [5, 10, 38, 57, 76, 82, 112, 122] [5, 16, 24, 26, 74, 77, 84, 89] [41, 44, 46, 53, 68, 91, 101, 127] [0, 47, 54, 63, 69, 77, 86, 113] [5, 7, 12, 42, 70, 99, 107, 125] [47, 55, 56, 58, 75, 80, 87, 118] [7, 13, 16, 63, 76, 95, 102, 109] [14, 17, 22, 45, 73, 82, 89, 116] [2, 50, 51, 56, 64, 78, 90, 122] [8, 9, 14, 16, 69, 79, 112, 127] [10, 17, 36, 48, 84, 86, 92, 119] [10, 19, 26, 33, 81, 87, 94, 98] [54, 55, 60, 61, 82, 90, 96, 123]
H_Z (53 checks, sparse supports)
[5, 22, 48, 55, 75, 82, 84, 107] [10, 14, 20, 60, 79, 82, 92, 94] [14, 21, 26, 57, 89, 98, 100, 112] [17, 24, 30, 62, 85, 89, 92, 103] [19, 27, 37, 42, 70, 94, 103, 121] [6, 13, 43, 49, 95, 104, 105, 108] [21, 35, 44, 61, 65, 67, 68, 123] [19, 24, 31, 36, 84, 98, 108, 110] [27, 33, 39, 63, 85, 95, 98, 113] [6, 29, 35, 47, 75, 100, 103, 113] [11, 15, 23, 41, 101, 104, 114, 115] [4, 31, 37, 45, 66, 67, 72, 73] [29, 33, 40, 46, 94, 101, 108, 117] [8, 35, 41, 43, 79, 91, 110, 117] [0, 15, 48, 58, 111, 113, 118, 119] [5, 37, 49, 63, 71, 76, 77, 125] [11, 38, 45, 54, 82, 105, 110, 113] [18, 25, 32, 51, 64, 111, 114, 120] [9, 40, 47, 52, 69, 71, 72, 80] [2, 20, 28, 57, 85, 115, 120, 122] [14, 41, 50, 58, 64, 73, 80, 127] [0, 26, 36, 59, 74, 81, 83, 86] [38, 43, 50, 53, 101, 103, 111, 122] [5, 15, 45, 51, 70, 89, 117, 122] [1, 20, 25, 55, 105, 118, 123, 124] [10, 13, 41, 47, 68, 76, 86, 87] [18, 48, 52, 59, 92, 105, 115, 117] [28, 34, 42, 56, 64, 65, 118, 125] [16, 50, 54, 57, 69, 74, 76, 90] [7, 21, 30, 36, 95, 119, 120, 125] [0, 24, 51, 55, 65, 77, 80, 90] [1, 2, 19, 46, 78, 81, 91, 93] [10, 25, 52, 56, 75, 98, 119, 122] [3, 30, 34, 60, 66, 115, 123, 126] [17, 23, 51, 54, 73, 78, 86, 96] [4, 27, 36, 44, 92, 101, 124, 126] [19, 28, 55, 59, 66, 87, 96, 114] [2, 6, 42, 61, 88, 90, 97, 99] [12, 31, 39, 46, 68, 104, 124, 125] [1, 33, 56, 60, 67, 83, 87, 90] [3, 7, 29, 53, 70, 88, 91, 102] [4, 8, 15, 63, 69, 93, 102, 106] [17, 34, 57, 61, 82, 108, 119, 124] [8, 39, 44, 62, 66, 71, 120, 127] [27, 32, 56, 59, 78, 80, 88, 106] [6, 9, 37, 46, 85, 111, 126, 127] [2, 21, 29, 60, 64, 71, 96, 106] [4, 7, 11, 35, 83, 97, 107, 109] [22, 40, 49, 53, 68, 73, 114, 126] [8, 12, 38, 58, 70, 75, 97, 112] [5, 9, 13, 25, 74, 99, 102, 112] [9, 12, 18, 45, 79, 93, 107, 116] [13, 20, 22, 50, 72, 78, 109, 116]
Code ID 128-22-10 · download JSON · raw on GitHub