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[[128,66,4]] d ≤
n
128
k
66
d
4
kd²/n
8.25
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · 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 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[41, 56, 77, 104]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[44, 69, 110, 111]
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 8 · H_Z 8 (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 1–2 (mean 1.938) · H_Z 1–2 (mean 1.938)
trapping sets H_X (1,1)×8 (2,1)×56 (3,1)×392 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 8 (1,2): 120 (2,1): 56 (2,2): 812 (3,1): 392 (3,2): 5488 (3,3): 168 (3,4): 1568
trapping sets H_Z (1,1)×8 (2,1)×56 (3,1)×392 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 8 (1,2): 120 (2,1): 56 (2,2): 812 (3,1): 392 (3,2): 5488 (3,3): 168 (3,4): 1568

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 67a4fc9c2444fbce9479181d (https://qecdb.org/codes/67a4fc9c2444fbce9479181d); 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: hyperbolic_2d; database distance claim d=4 (lower 4, upper 4).
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 topological (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

[[128,66,4]] — qecdb.org hyperbolic_2d 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 67a4fc9c2444fbce9479181d.

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: [[128,66,4]] at w = 8, record 67a4fc9c2444fbce9479181d; qecdb description

    • "hyperbolic_2d", database distance claim d = 4 (lower 4, upper 4).

Evidence trail

  • Ladder: 4 at deep → 4 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [34, 38, 113, 120] and Z witness [42, 46, 89, 98], both of weight 4.

  • Claim, precisely: a witness-backed upper bound d ≤ 4. 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 6 entries on d, k, n: 128-64-2.json, 136-52-2.json, 152-58-2.json, 162-54-3.json, 168-48-3.json, 176-46-4.json. Efficiency k d² / n = 8.25.
  • 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/67a4fc9c2444fbce9479181d (the API path /api/codes/67a4fc9c2444fbce9479181d) 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 31 (max weight 8) · Z-checks 31 (max weight 8)
H_X (31 checks, sparse supports)
[21, 52, 85, 86, 95, 108, 114, 119] [32, 33, 50, 92, 94, 96, 104, 121] [6, 11, 15, 18, 30, 58, 73, 91] [8, 14, 16, 20, 43, 93, 124, 126] [1, 26, 43, 45, 46, 53, 98, 108] [5, 8, 9, 11, 29, 35, 79, 121] [4, 9, 13, 31, 53, 56, 80, 105] [15, 39, 72, 74, 78, 82, 96, 122] [4, 23, 27, 48, 60, 86, 117, 126] [7, 22, 39, 55, 67, 69, 77, 111] [0, 37, 45, 60, 65, 81, 106, 112] [10, 52, 63, 67, 71, 106, 115, 125] [18, 22, 24, 49, 61, 90, 94, 97] [44, 49, 63, 72, 110, 113, 118, 120] [2, 36, 83, 97, 101, 122, 123, 125] [1, 5, 25, 48, 57, 100, 102, 103] [42, 56, 68, 89, 93, 95, 100, 112] [13, 16, 37, 38, 57, 70, 85, 107] [34, 38, 42, 47, 71, 87, 98, 117] [0, 2, 34, 69, 75, 76, 110, 119] [10, 19, 26, 27, 59, 68, 75, 107] [7, 19, 65, 66, 87, 114, 118, 123] [3, 28, 61, 64, 66, 76, 82, 115] [3, 17, 44, 50, 54, 73, 101, 111] [35, 40, 51, 54, 78, 88, 90, 109] [14, 30, 84, 92, 103, 105, 109, 116] [17, 24, 41, 62, 74, 79, 99, 116] [6, 28, 32, 36, 77, 88, 99, 113] [25, 51, 58, 62, 80, 104, 124, 127] [12, 20, 31, 33, 40, 41, 91, 102] [12, 23, 29, 46, 70, 84, 89, 127]
H_Z (31 checks, sparse supports)
[19, 24, 26, 53, 61, 66, 105, 116] [8, 22, 24, 52, 67, 79, 86, 126] [5, 37, 39, 55, 57, 81, 96, 121] [34, 51, 88, 110, 113, 117, 124, 126] [11, 15, 21, 29, 70, 83, 85, 122] [42, 56, 80, 87, 96, 104, 122, 123] [4, 27, 28, 62, 75, 76, 80, 99] [1, 6, 26, 59, 91, 102, 113, 120] [12, 28, 40, 45, 46, 64, 81, 88] [51, 65, 66, 78, 82, 89, 112, 127] [8, 16, 35, 38, 47, 72, 78, 120] [23, 29, 35, 60, 61, 90, 106, 115] [5, 10, 11, 18, 27, 48, 49, 63] [4, 9, 71, 94, 97, 117, 121, 125] [7, 39, 62, 74, 93, 95, 114, 124] [14, 16, 17, 69, 85, 111, 116, 119] [2, 13, 34, 38, 50, 92, 101, 105] [31, 32, 33, 36, 47, 53, 83, 98] [18, 30, 46, 84, 97, 108, 114, 123] [1, 7, 22, 45, 65, 92, 94, 103] [12, 52, 73, 89, 91, 95, 101, 125] [33, 50, 67, 100, 102, 106, 111, 112] [0, 25, 32, 48, 60, 69, 77, 104] [14, 43, 49, 87, 90, 98, 109, 118] [15, 19, 25, 58, 68, 72, 100, 118] [2, 6, 23, 36, 58, 86, 119, 127] [3, 10, 17, 31, 41, 56, 68, 115] [20, 21, 41, 43, 55, 77, 99, 108] [9, 13, 59, 64, 74, 79, 82, 107] [0, 3, 37, 54, 70, 76, 84, 109] [20, 40, 42, 44, 54, 63, 71, 93]
Code ID 128-66-4 · download JSON · raw on GitHub