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[[116,2,15]] d ≤
n
116
k
2
d
15
kd²/n
3.879
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[12, 25, 29, 38, 41, 43, 48, 53, 61, 66, 72, 81, 90, 93, 100]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[17, 20, 29, 38, 44, 60, 62, 69, 73, 86, 94, 99, 107, 112, 115]
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 6 · H_Z 6
qubit degrees H_X 1–4 (mean 2.948) · H_Z 1–4 (mean 2.948)
trapping sets H_X (1,1)×2 (2,1)×2 (3,1)×2 (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): 56 (1,3): 4 (1,4): 54 (2,1): 2 (2,2): 55 (2,3): 32 (2,4): 424 (2,5): 36 (2,6): 306 (3,1): 2 (3,2): 54 (3,3): 136 (3,4): 1772 (3,5): 542 (3,6): 4568 (3,7): 396 (3,8): 2472 (3,9): 36 (3,10): 192
trapping sets H_Z (1,1)×2 (2,1)×2 (3,1)×2 (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): 56 (1,3): 4 (1,4): 54 (2,1): 2 (2,2): 55 (2,3): 32 (2,4): 424 (2,5): 36 (2,6): 306 (3,1): 2 (3,2): 54 (3,3): 136 (3,4): 1772 (3,5): 542 (3,6): 4568 (3,7): 396 (3,8): 2472 (3,9): 36 (3,10): 192

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 6a9e4f39a1dc9c110d5c4e63 (https://qecdb.org/codes/6a9e4f39a1dc9c110d5c4e63); 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: bivariate bicycle; database distance claim d=15 (lower 15, upper 15).
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. Filed as `codes/116-2-15-b.json`, a lighter-check companion to `codes/116-2-15.json` (same [[116,2,15]], max check weight 6 vs 8).
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[116,2,15]] — qecdb.org bivariate bicycle code, reconstructed from its H field

Direction & hypothesis

Track cell weight-6 × 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 6a9e4f39a1dc9c110d5c4e63.

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: [[116,2,15]] at w = 6, record 6a9e4f39a1dc9c110d5c4e63; qecdb description

    • "bivariate bicycle", database distance claim d = 15 (lower 15, upper 15).

Evidence trail

  • Ladder: 15 at deep2 → 15 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [3, 9, 11, 16, 24, 27, 37, 50, 66, 67, 94, 98, 103, 106, 112] and Z witness [0, 7, 9, 28, 49, 64, 69, 77, 82, 89, 95, 96, 98, 102, 109], both of weight 15.

  • Claim, precisely: a witness-backed upper bound d ≤ 15. 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: 118-2-15.json. Efficiency k d² / n = 3.88.
  • Verdict: passed: true, not refuted, not a duplicate of a board entry;

    • labels: advances the weight-6 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/6a9e4f39a1dc9c110d5c4e63 (the API path /api/codes/6a9e4f39a1dc9c110d5c4e63) 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 57 (max weight 6) · Z-checks 57 (max weight 6)
H_X (57 checks, sparse supports)
[0, 9, 21, 40, 58, 71] [1, 10, 22, 41, 59, 72] [2, 11, 23, 42, 60, 73] [3, 12, 24, 43, 61, 74] [4, 13, 25, 44, 62, 75] [5, 14, 26, 45, 63, 76] [6, 15, 27, 46, 64, 77] [7, 16, 28, 47, 65, 78] [8, 17, 29, 48, 66, 79] [9, 18, 30, 49, 67, 80] [10, 19, 31, 50, 68, 81] [11, 20, 32, 51, 69, 82] [12, 21, 33, 52, 70, 83] [13, 22, 34, 53, 71, 84] [14, 23, 35, 54, 72, 85] [15, 24, 36, 55, 73, 86] [16, 25, 37, 56, 74, 87] [17, 26, 38, 57, 75, 88] [0, 18, 27, 39, 76, 89] [1, 19, 28, 40, 77, 90] [2, 20, 29, 41, 78, 91] [3, 21, 30, 42, 79, 92] [4, 22, 31, 43, 80, 93] [5, 23, 32, 44, 81, 94] [6, 24, 33, 45, 82, 95] [7, 25, 34, 46, 83, 96] [8, 26, 35, 47, 84, 97] [9, 27, 36, 48, 85, 98] [10, 28, 37, 49, 86, 99] [11, 29, 38, 50, 87, 100] [12, 30, 39, 51, 88, 101] [13, 31, 40, 52, 89, 102] [14, 32, 41, 53, 90, 103] [15, 33, 42, 54, 91, 104] [16, 34, 43, 55, 92, 105] [17, 35, 44, 56, 93, 106] [18, 36, 45, 57, 94, 107] [0, 19, 37, 46, 95, 108] [1, 20, 38, 47, 96, 109] [2, 21, 39, 48, 97, 110] [3, 22, 40, 49, 98, 111] [4, 23, 41, 50, 99, 112] [5, 24, 42, 51, 100, 113] [6, 25, 43, 52, 101, 114] [7, 26, 44, 53, 102, 115] [8, 27, 45, 54, 58, 103] [9, 28, 46, 55, 59, 104] [10, 29, 47, 56, 60, 105] [11, 30, 48, 57, 61, 106] [0, 12, 31, 49, 62, 107] [1, 13, 32, 50, 63, 108] [2, 14, 33, 51, 64, 109] [3, 15, 34, 52, 65, 110] [4, 16, 35, 53, 66, 111] [5, 17, 36, 54, 67, 112] [6, 18, 37, 55, 68, 113] [7, 19, 38, 56, 69, 114]
H_Z (57 checks, sparse supports)
[0, 45, 58, 76, 95, 107] [1, 46, 59, 77, 96, 108] [2, 47, 60, 78, 97, 109] [3, 48, 61, 79, 98, 110] [4, 49, 62, 80, 99, 111] [5, 50, 63, 81, 100, 112] [6, 51, 64, 82, 101, 113] [7, 52, 65, 83, 102, 114] [8, 53, 66, 84, 103, 115] [9, 54, 58, 67, 85, 104] [10, 55, 59, 68, 86, 105] [11, 56, 60, 69, 87, 106] [12, 57, 61, 70, 88, 107] [0, 13, 62, 71, 89, 108] [1, 14, 63, 72, 90, 109] [2, 15, 64, 73, 91, 110] [3, 16, 65, 74, 92, 111] [4, 17, 66, 75, 93, 112] [5, 18, 67, 76, 94, 113] [6, 19, 68, 77, 95, 114] [7, 20, 69, 78, 96, 115] [8, 21, 58, 70, 79, 97] [9, 22, 59, 71, 80, 98] [10, 23, 60, 72, 81, 99] [11, 24, 61, 73, 82, 100] [12, 25, 62, 74, 83, 101] [13, 26, 63, 75, 84, 102] [14, 27, 64, 76, 85, 103] [15, 28, 65, 77, 86, 104] [16, 29, 66, 78, 87, 105] [17, 30, 67, 79, 88, 106] [18, 31, 68, 80, 89, 107] [19, 32, 69, 81, 90, 108] [20, 33, 70, 82, 91, 109] [21, 34, 71, 83, 92, 110] [22, 35, 72, 84, 93, 111] [23, 36, 73, 85, 94, 112] [24, 37, 74, 86, 95, 113] [25, 38, 75, 87, 96, 114] [26, 39, 76, 88, 97, 115] [27, 40, 58, 77, 89, 98] [28, 41, 59, 78, 90, 99] [29, 42, 60, 79, 91, 100] [30, 43, 61, 80, 92, 101] [31, 44, 62, 81, 93, 102] [32, 45, 63, 82, 94, 103] [33, 46, 64, 83, 95, 104] [34, 47, 65, 84, 96, 105] [35, 48, 66, 85, 97, 106] [36, 49, 67, 86, 98, 107] [37, 50, 68, 87, 99, 108] [38, 51, 69, 88, 100, 109] [39, 52, 70, 89, 101, 110] [40, 53, 71, 90, 102, 111] [41, 54, 72, 91, 103, 112] [42, 55, 73, 92, 104, 113] [43, 56, 74, 93, 105, 114]
Code ID 116-2-15-b · download JSON · raw on GitHub