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[[84,6,6]] d ≤
n
84
k
6
d
6
kd²/n
2.571
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[47, 56, 57, 66, 67, 78]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[0, 1, 8, 16, 17, 40]
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 5 · H_Z 5
qubit degrees H_X 1–3 (mean 2.321) · H_Z 1–3 (mean 2.321)
trapping sets H_X (1,1)×6 (2,1)×8 (3,1)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 6 (1,2): 45 (1,3): 33 (2,1): 8 (2,2): 84 (2,3): 216 (2,4): 82 (3,1): 14 (3,2): 206 (3,3): 923 (3,4): 930 (3,5): 301 (3,6): 166 (3,7): 22
trapping sets H_Z (1,1)×6 (2,1)×6 (3,1)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 6 (1,2): 45 (1,3): 33 (2,1): 6 (2,2): 90 (2,3): 210 (2,4): 84 (3,1): 14 (3,2): 218 (3,3): 904 (3,4): 936 (3,5): 304 (3,6): 162 (3,7): 24

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 67bf07bae8112da5fce0c04d (https://qecdb.org/codes/67bf07bae8112da5fce0c04d); 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=6 (lower 6, upper 6).
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 ≤ 6 (computed)

How this code was found

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

[[84,6,6]] — qecdb.org 2BGA 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 67bf07bae8112da5fce0c04d.

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: [[84,6,6]] at w = 5, record 67bf07bae8112da5fce0c04d; qecdb description

    • "2BGA", database distance claim d = 6 (lower 6, upper 6).

Evidence trail

  • Ladder: 6 at deep2 → 6 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [49, 58, 59, 65, 69, 74] and Z witness [3, 6, 11, 12, 24, 25], both of weight 6.

  • Claim, precisely: a witness-backed upper bound d ≤ 6. 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 3 entries on n, w: 108-6-6.json, 88-6-6.json, 96-6-6.json. Efficiency k d² / n = 2.57.
  • 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/67bf07bae8112da5fce0c04d (the API path /api/codes/67bf07bae8112da5fce0c04d) 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 39 (max weight 5) · Z-checks 39 (max weight 5)
H_X (39 checks, sparse supports)
[9, 21, 67, 73, 83] [35, 41, 51, 66, 68] [18, 32, 51, 64, 82] [20, 31, 74, 76, 83] [15, 28, 67, 69, 70] [8, 17, 64, 68, 79] [4, 32, 46, 52, 80] [29, 34, 50, 67, 79] [13, 18, 48, 62, 77] [7, 15, 58, 63, 78] [27, 33, 50, 70, 83] [26, 30, 51, 69, 79] [13, 27, 44, 55, 81] [22, 36, 64, 66, 74] [16, 40, 70, 73, 76] [1, 40, 42, 53, 81] [37, 39, 47, 52, 53] [34, 38, 44, 45, 56] [14, 26, 45, 46, 49] [10, 41, 43, 46, 47] [7, 22, 47, 65, 80] [11, 25, 42, 52, 65] [3, 6, 48, 59, 61] [19, 37, 66, 76, 82] [1, 17, 60, 71, 77] [0, 16, 61, 63, 75] [2, 35, 57, 62, 71] [29, 39, 54, 60, 78] [11, 12, 54, 58, 71] [2, 9, 42, 55, 56] [23, 28, 49, 56, 81] [10, 21, 61, 72, 78] [19, 38, 48, 57, 75] [5, 31, 53, 55, 65] [5, 30, 58, 60, 62] [3, 24, 43, 44, 49] [0, 8, 43, 45, 80] [14, 20, 59, 72, 75] [6, 25, 73, 74, 82]
H_Z (39 checks, sparse supports)
[15, 21, 29, 67, 78] [19, 35, 36, 57, 66] [12, 17, 35, 68, 71] [17, 18, 36, 64, 77] [6, 18, 19, 48, 82] [16, 19, 20, 75, 76] [8, 22, 32, 64, 80] [18, 30, 35, 51, 62] [7, 10, 39, 47, 78] [1, 2, 11, 42, 71] [6, 16, 21, 61, 73] [15, 16, 33, 63, 70] [27, 28, 40, 70, 81] [12, 29, 33, 50, 54] [17, 29, 30, 60, 79] [26, 32, 41, 46, 51] [9, 28, 34, 56, 67] [22, 37, 41, 47, 66] [24, 27, 34, 44, 50] [22, 25, 31, 65, 74] [8, 26, 34, 45, 79] [9, 25, 40, 42, 73] [2, 5, 13, 55, 62] [20, 21, 33, 72, 83] [1, 13, 23, 77, 81] [0, 4, 7, 63, 80] [1, 5, 39, 53, 60] [4, 11, 39, 52, 54] [4, 10, 14, 46, 72] [24, 26, 28, 49, 69] [31, 37, 40, 53, 76] [0, 14, 38, 45, 75] [3, 13, 38, 44, 48] [25, 32, 37, 52, 82] [3, 14, 23, 49, 59] [8, 24, 41, 43, 68] [9, 27, 31, 55, 83] [5, 7, 11, 58, 65] [6, 20, 36, 59, 74]
Code ID 84-6-6 · download JSON · raw on GitHub