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[[60,20,5]] d ≤
n
60
k
20
d
5
kd²/n
8.333
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · 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 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[1, 7, 11, 13, 15]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[1, 7, 11, 13, 15]
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 4 · H_Z 4 (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 2.667) · H_Z 2–4 (mean 2.667)
trapping sets H_X (1,2)×26 (2,1)×20 (3,1)×80 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 26 (1,3): 28 (1,4): 6 (2,1): 20 (2,2): 105 (2,3): 190 (2,4): 116 (2,5): 22 (3,1): 80 (3,2): 442 (3,3): 1163 (3,4): 1396 (3,5): 791 (3,6): 224 (3,7): 40 (3,8): 6
trapping sets H_Z (1,2)×26 (2,1)×20 (3,1)×80 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 26 (1,3): 28 (1,4): 6 (2,1): 20 (2,2): 105 (2,3): 190 (2,4): 116 (2,5): 22 (3,1): 80 (3,2): 442 (3,3): 1163 (3,4): 1396 (3,5): 791 (3,6): 224 (3,7): 40 (3,8): 6

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Reconstructed from qecdb.org record 6798877b5a99a6b92bb92002 (https://qecdb.org/codes/6798877b5a99a6b92bb92002); 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: none; database distance claim d=5 (lower 5, upper 5).
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 other (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

[[60,20,5]] — qecdb.org none 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 6798877b5a99a6b92bb92002.

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: [[60,20,5]] at w = 8, record 6798877b5a99a6b92bb92002; qecdb description

    • "none", database distance claim d = 5 (lower 5, upper 5).

Evidence trail

  • Ladder: 5 at deep → 5 at screen1

    • → packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [29, 32, 40, 49, 53] and Z witness [2, 5, 34, 51, 56], both of weight 5.

  • Claim, precisely: a witness-backed upper bound d ≤ 5. 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: 65-17-3.json, 80-20-5.json. Efficiency k d² / n = 8.33.
  • 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/6798877b5a99a6b92bb92002 (the API path /api/codes/6798877b5a99a6b92bb92002) 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 20 (max weight 8) · Z-checks 20 (max weight 8)
H_X (20 checks, sparse supports)
[2, 19, 21, 22, 28, 34, 47, 54] [2, 5, 22, 47, 51, 54, 56, 59] [4, 6, 9, 16, 37, 46, 48, 55] [3, 25, 27, 39, 43, 49, 53, 58] [0, 3, 17, 27, 39, 42, 50, 58] [14, 24, 28, 32, 38, 40, 54, 59] [0, 3, 5, 6, 25, 44, 46, 51] [1, 4, 9, 13, 20, 44, 46, 55] [10, 26, 29, 30, 42, 45, 49, 58] [14, 21, 23, 24, 36, 41, 54, 59] [1, 7, 8, 20, 31, 35, 52, 57] [4, 11, 12, 13, 14, 23, 37, 38] [27, 31, 33, 36, 41, 43, 50, 57] [7, 15, 22, 26, 30, 34, 35, 56] [24, 31, 32, 33, 41, 43, 52, 53] [2, 3, 5, 16, 19, 39, 44, 46] [8, 9, 18, 20, 33, 48, 52, 57] [10, 11, 12, 15, 17, 26, 42, 45] [4, 10, 12, 14, 23, 29, 40, 55] [8, 18, 26, 34, 35, 45, 47, 56]
H_Z (20 checks, sparse supports)
[2, 19, 21, 22, 28, 34, 47, 54] [2, 5, 22, 47, 51, 54, 56, 59] [4, 6, 9, 16, 37, 46, 48, 55] [3, 25, 27, 39, 43, 49, 53, 58] [0, 3, 17, 27, 39, 42, 50, 58] [14, 24, 28, 32, 38, 40, 54, 59] [0, 3, 5, 6, 25, 44, 46, 51] [1, 4, 9, 13, 20, 44, 46, 55] [10, 26, 29, 30, 42, 45, 49, 58] [14, 21, 23, 24, 36, 41, 54, 59] [1, 7, 8, 20, 31, 35, 52, 57] [4, 11, 12, 13, 14, 23, 37, 38] [27, 31, 33, 36, 41, 43, 50, 57] [7, 15, 22, 26, 30, 34, 35, 56] [24, 31, 32, 33, 41, 43, 52, 53] [2, 3, 5, 16, 19, 39, 44, 46] [8, 9, 18, 20, 33, 48, 52, 57] [10, 11, 12, 15, 17, 26, 42, 45] [4, 10, 12, 14, 23, 29, 40, 55] [8, 18, 26, 34, 35, 45, 47, 56]
Code ID 60-20-5 · download JSON · raw on GitHub