[[112,20,9]] · QEC Challenge

[[112,20,9]] d ≤

112

kd²/n

14.464

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[16, 47, 51, 72, 73, 87, 89, 102, 111]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[0, 10, 11, 13, 36, 39, 67, 86, 108]

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 8 · H_Z 8

qubit degrees H_X 2–4 (mean 3.286) · H_Z 2–4 (mean 3.286)

trapping sets H_X (1,2)×11 (2,2)×7 (3,2)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 11 (1,3): 58 (1,4): 43 (2,2): 7 (2,3): 50 (2,4): 377 (2,5): 594 (2,6): 260 (3,2): 12 (3,3): 113 (3,4): 784 (3,5): 3565 (3,6): 7287 (3,7): 6801 (3,8): 2959 (3,9): 795 (3,10): 205

trapping sets H_Z (1,2)×11 (2,2)×2 (3,2)×8 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,2): 11 (1,3): 58 (1,4): 43 (2,2): 2 (2,3): 66 (2,4): 359 (2,5): 602 (2,6): 259 (3,2): 8 (3,3): 109 (3,4): 859 (3,5): 3478 (3,6): 7259 (3,7): 6887 (3,8): 2898 (3,9): 822 (3,10): 201

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction Reconstructed from qecdb.org record 6a41415955928fd7347649ac (https://qecdb.org/codes/6a41415955928fd7347649ac); 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 w=8 wa=4 wb=4 G=SmallGroup(56, 8); arXiv:2306.16400; database distance claim d=9 (lower None, upper None).

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 ≤ 8 (computed)

How this code was found

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

[[112,20,9]] — qecdb.org 2BGA w=8 wa=4 wb=4 G=SmallGroup(56, 8); arXiv:2306.16400 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 6a41415955928fd7347649ac.

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: [[112,20,9]] at w = 8, record 6a41415955928fd7347649ac; qecdb description

"2BGA w=8 wa=4 wb=4 G=SmallGroup(56, 8); arXiv:2306.16400", database distance claim d = ? (lower ?, upper ?).

Evidence trail

Ladder: 9 at deep2 → 9 at screen1

→ packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [16, 47, 51, 72, 73, 87, 89, 102, 111] and Z witness [3, 12, 28, 29, 40, 43, 85, 98, 105], both of weight 9.

Claim, precisely: a witness-backed upper bound d ≤ 9. 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: 144-18-9.json. Efficiency k d² / n = 14.46.
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/6a41415955928fd7347649ac (the API path /api/codes/6a41415955928fd7347649ac) 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 46 (max weight 8) · Z-checks 46 (max weight 8)

H_X (46 checks, sparse supports)

[25, 36, 50, 52, 67, 80, 92, 103] [29, 40, 41, 53, 71, 84, 90, 97] [19, 30, 47, 54, 73, 74, 86, 108] [4, 33, 44, 55, 75, 88, 100, 109] [16, 42, 44, 45, 59, 89, 94, 101] [23, 34, 35, 55, 77, 78, 91, 96] [7, 37, 48, 49, 79, 92, 98, 105] [20, 32, 46, 49, 62, 88, 93, 99] [9, 27, 38, 53, 81, 82, 94, 111] [10, 38, 39, 48, 64, 83, 95, 100] [11, 13, 41, 51, 63, 83, 96, 107] [0, 24, 51, 52, 66, 97, 102, 108] [13, 31, 42, 43, 85, 86, 99, 104] [14, 26, 43, 51, 68, 82, 87, 105] [2, 15, 45, 54, 56, 87, 100, 110] [1, 28, 40, 54, 70, 96, 101, 106] [7, 17, 35, 46, 69, 89, 90, 102] [2, 18, 46, 47, 72, 91, 103, 107] [5, 19, 21, 49, 61, 71, 91, 104] [3, 5, 32, 55, 57, 74, 105, 111] [0, 21, 39, 50, 58, 93, 94, 106] [5, 22, 34, 50, 76, 90, 95, 110] [8, 9, 23, 52, 59, 65, 95, 107] [6, 9, 36, 48, 60, 78, 104, 108] [1, 15, 25, 43, 57, 77, 97, 98] [1, 8, 26, 53, 61, 80, 99, 109] [12, 27, 29, 54, 58, 68, 79, 99] [10, 12, 13, 40, 62, 69, 82, 110] [3, 4, 29, 47, 60, 64, 101, 102] [4, 12, 30, 42, 65, 84, 98, 103] [16, 17, 31, 55, 61, 66, 73, 103] [2, 14, 17, 44, 58, 67, 86, 111] [6, 23, 33, 50, 56, 62, 85, 105] [6, 7, 16, 34, 63, 68, 88, 106] [9, 20, 35, 37, 64, 76, 87, 106] [18, 20, 21, 48, 65, 70, 77, 90] [10, 11, 37, 53, 57, 67, 72, 108] [13, 24, 25, 39, 68, 74, 81, 109] [8, 22, 25, 51, 64, 69, 75, 94] [4, 14, 31, 41, 59, 70, 93, 110] [14, 15, 24, 42, 60, 71, 76, 96] [17, 28, 43, 45, 60, 72, 84, 95] [2, 26, 28, 29, 73, 78, 85, 98] [7, 18, 19, 45, 62, 75, 80, 111] [21, 32, 33, 47, 63, 76, 82, 89] [11, 22, 39, 49, 65, 66, 78, 101]

H_Z (46 checks, sparse supports)

[14, 32, 37, 50, 67, 76, 93, 105] [19, 24, 36, 53, 71, 80, 81, 108] [20, 26, 31, 54, 70, 73, 87, 99] [4, 22, 40, 45, 75, 84, 101, 110] [23, 28, 41, 42, 59, 84, 85, 96] [18, 25, 30, 55, 74, 75, 77, 103] [7, 27, 32, 44, 79, 88, 89, 111] [10, 33, 45, 46, 62, 72, 89, 100] [9, 28, 34, 39, 78, 81, 95, 106] [22, 29, 35, 48, 64, 78, 79, 90] [11, 30, 48, 52, 65, 83, 92, 108] [0, 31, 36, 49, 66, 92, 93, 104] [13, 26, 33, 38, 82, 83, 85, 109] [16, 27, 39, 51, 66, 68, 83, 94] [15, 35, 40, 51, 69, 87, 96, 97] [1, 18, 41, 52, 70, 80, 97, 107] [17, 36, 42, 47, 60, 86, 89, 103] [2, 30, 37, 43, 72, 86, 87, 98] [2, 19, 38, 55, 73, 91, 100, 111] [3, 39, 44, 54, 58, 74, 100, 101] [21, 34, 41, 46, 63, 90, 91, 93] [5, 24, 35, 47, 74, 76, 91, 102] [5, 23, 43, 48, 77, 95, 104, 105] [6, 26, 49, 55, 61, 78, 88, 105] [0, 25, 44, 53, 67, 94, 97, 109] [8, 38, 45, 50, 56, 80, 94, 95] [8, 13, 27, 46, 69, 81, 99, 107] [9, 10, 47, 51, 64, 82, 107, 108] [1, 29, 42, 49, 71, 98, 99, 101] [12, 32, 43, 53, 57, 82, 84, 99] [2, 12, 31, 50, 58, 85, 103, 110] [13, 14, 34, 54, 68, 86, 96, 110] [3, 7, 33, 51, 63, 75, 102, 105] [4, 16, 46, 52, 59, 88, 102, 103] [1, 16, 21, 35, 61, 77, 89, 106] [5, 17, 18, 53, 61, 72, 90, 111] [0, 6, 37, 54, 56, 79, 106, 108] [4, 8, 20, 39, 64, 65, 93, 109] [9, 21, 22, 42, 65, 76, 94, 104] [5, 10, 15, 41, 57, 71, 83, 110] [1, 11, 24, 55, 57, 66, 96, 109] [6, 24, 29, 43, 60, 68, 85, 97] [7, 12, 25, 26, 68, 69, 80, 98] [3, 9, 14, 45, 59, 60, 87, 111] [11, 16, 28, 47, 63, 72, 73, 101] [12, 18, 23, 49, 62, 65, 79, 91]

Code ID 112-20-9 · download JSON · raw on GitHub