[[15,5,3]] — qecdb.org hyperbolic_2d 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 67a4b0119edf81e4b7e670ac.

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: [[15,5,3]] at w = 5, record 67a4b0119edf81e4b7e670ac; qecdb description

"hyperbolic_2d", database distance claim d = 3 (lower 3, upper 3).

Evidence trail

• Ladder: 3 at deep2 → 3 at screen1

→ packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [0, 12, 13] and Z witness [1, 2, 3], both of weight 3.

• Claim, precisely: a witness-backed upper bound d ≤ 3. 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 k, n: 18-4-3.json. Efficiency k d² / n = 3.00.
• 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/67a4b0119edf81e4b7e670ac (the API path /api/codes/67a4b0119edf81e4b7e670ac) 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.