[[126,26,9]] · QEC Challenge

[[126,26,9]] d ≤

126

kd²/n

16.714

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, 18, 22, 64, 66, 74, 97, 113, 118]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[7, 26, 48, 59, 61, 62, 97, 113, 118]

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 1–4 (mean 3.175) · H_Z 1–4 (mean 3.175)

trapping sets H_X (1,1)×4 (2,2)×22 (3,1)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 14 (1,3): 64 (1,4): 44 (2,2): 22 (2,3): 109 (2,4): 416 (2,5): 573 (2,6): 280 (3,1): 2 (3,2): 39 (3,3): 345 (3,4): 1345 (3,5): 4220 (3,6): 7206 (3,7): 6597 (3,8): 3040 (3,9): 760 (3,10): 258

trapping sets H_Z (1,1)×4 (2,1)×1 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 16 (1,3): 60 (1,4): 46 (2,1): 1 (2,2): 20 (2,3): 121 (2,4): 392 (2,5): 560 (2,6): 306 (3,1): 1 (3,2): 46 (3,3): 350 (3,4): 1383 (3,5): 4025 (3,6): 7017 (3,7): 6736 (3,8): 3262 (3,9): 758 (3,10): 304

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction Reconstructed from qecdb.org record 6a422a12d15d481af09df8a4 (https://qecdb.org/codes/6a422a12d15d481af09df8a4); 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(63, 2); 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

[[126,26,9]] — qecdb.org 2BGA w=8 wa=4 wb=4 G=SmallGroup(63, 2); 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 6a422a12d15d481af09df8a4.

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: [[126,26,9]] at w = 8, record 6a422a12d15d481af09df8a4; qecdb description

"2BGA w=8 wa=4 wb=4 G=SmallGroup(63, 2); arXiv:2306.16400", database distance claim d = ? (lower ?, upper ?).

Evidence trail

Ladder: 9 at deep → 9 at screen1

→ packaging witness at 8000 trials/side (numpy lightest_logical, seed 0): X witness [45, 54, 63, 66, 69, 77, 78, 104, 110] and Z witness [11, 21, 51, 72, 74, 76, 96, 114, 125], 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 2 entries on d, k, n: 128-21-8.json, 144-18-9.json. Efficiency k d² / n = 16.71.
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/6a422a12d15d481af09df8a4 (the API path /api/codes/6a422a12d15d481af09df8a4) 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 50 (max weight 8) · Z-checks 50 (max weight 8)

H_X (50 checks, sparse supports)

[12, 32, 43, 54, 78, 99, 107, 110] [18, 38, 49, 52, 78, 84, 97, 113] [21, 41, 51, 59, 87, 108, 115, 118] [12, 22, 52, 60, 87, 88, 90, 109] [7, 16, 54, 57, 84, 90, 103, 118] [27, 47, 56, 58, 87, 93, 106, 120] [18, 28, 33, 57, 88, 93, 94, 107] [3, 11, 30, 50, 96, 116, 121, 123] [21, 31, 58, 62, 96, 97, 99, 117] [22, 32, 33, 45, 97, 98, 100, 118] [15, 25, 59, 61, 93, 99, 112, 123] [16, 26, 39, 60, 94, 99, 100, 113] [6, 9, 36, 55, 96, 102, 114, 124] [27, 37, 42, 61, 97, 102, 103, 115] [28, 38, 39, 43, 87, 98, 103, 104] [8, 19, 39, 57, 67, 72, 105, 125] [9, 20, 30, 40, 105, 106, 108, 122] [31, 41, 42, 53, 106, 107, 109, 123] [11, 14, 24, 34, 102, 108, 119, 125] [25, 35, 48, 62, 103, 108, 109, 120] [13, 17, 45, 60, 66, 77, 105, 111] [0, 14, 36, 46, 106, 111, 112, 121] [37, 47, 48, 51, 96, 107, 112, 113] [16, 28, 48, 61, 63, 73, 80, 83] [17, 29, 39, 49, 63, 74, 114, 116] [0, 4, 40, 50, 114, 115, 117, 125] [19, 23, 33, 43, 69, 83, 111, 116] [1, 20, 34, 44, 112, 116, 117, 124] [22, 26, 53, 62, 63, 64, 71, 86] [2, 23, 45, 54, 64, 72, 114, 119] [1, 3, 46, 55, 105, 115, 119, 120] [1, 14, 25, 37, 65, 81, 89, 92] [26, 38, 48, 56, 65, 66, 67, 82] [2, 10, 49, 57, 66, 83, 121, 122] [28, 32, 42, 51, 64, 67, 76, 92] [5, 29, 43, 52, 67, 77, 119, 122] [4, 20, 31, 35, 65, 68, 79, 95] [7, 32, 53, 59, 66, 68, 69, 80] [5, 8, 54, 60, 63, 69, 121, 124] [5, 23, 34, 46, 70, 90, 98, 101] [1, 6, 35, 47, 70, 71, 73, 91] [7, 18, 56, 61, 71, 72, 74, 92] [0, 3, 37, 41, 68, 73, 85, 101] [12, 38, 51, 58, 69, 73, 74, 86] [10, 29, 40, 44, 70, 75, 88, 104] [4, 11, 15, 41, 71, 75, 76, 89] [12, 16, 59, 62, 65, 72, 76, 77] [5, 13, 44, 55, 78, 79, 81, 100] [6, 14, 15, 27, 79, 80, 82, 101] [2, 8, 46, 50, 75, 81, 94, 110]

H_Z (50 checks, sparse supports)

[24, 25, 30, 41, 85, 89, 108, 123] [30, 31, 36, 47, 91, 95, 96, 106] [33, 34, 39, 50, 94, 98, 116, 125] [21, 34, 35, 40, 70, 95, 108, 117] [15, 34, 36, 37, 89, 101, 102, 112] [39, 40, 45, 55, 100, 104, 105, 114] [27, 40, 41, 46, 75, 101, 106, 115] [42, 43, 48, 57, 67, 83, 103, 107] [30, 43, 44, 49, 78, 104, 116, 122] [15, 31, 44, 50, 75, 79, 117, 123] [24, 43, 45, 46, 98, 110, 111, 119] [25, 44, 46, 47, 70, 81, 112, 120] [48, 49, 53, 60, 63, 66, 109, 113] [36, 49, 50, 54, 84, 110, 114, 121] [21, 37, 50, 55, 81, 85, 96, 115] [0, 1, 51, 61, 73, 92, 112, 115] [39, 51, 52, 56, 67, 74, 87, 113] [24, 40, 52, 57, 84, 88, 122, 125] [33, 51, 53, 54, 64, 69, 107, 118] [34, 52, 54, 55, 78, 90, 119, 124] [1, 4, 56, 62, 65, 71, 117, 120] [45, 56, 57, 59, 66, 72, 93, 118] [30, 46, 57, 60, 90, 94, 105, 121] [2, 3, 5, 14, 81, 101, 119, 121] [3, 6, 48, 58, 73, 82, 96, 120] [33, 49, 58, 61, 74, 83, 93, 97] [3, 4, 42, 59, 68, 76, 115, 123] [43, 58, 59, 60, 69, 77, 87, 99] [5, 6, 10, 20, 70, 79, 122, 124] [6, 11, 53, 61, 71, 80, 102, 123] [39, 54, 61, 62, 63, 72, 99, 103] [7, 8, 12, 23, 69, 72, 90, 110] [1, 8, 9, 13, 81, 91, 105, 124] [9, 14, 42, 56, 82, 92, 102, 106] [0, 8, 10, 11, 75, 85, 121, 125] [9, 11, 51, 62, 76, 86, 96, 108] [12, 13, 18, 29, 74, 77, 78, 88] [4, 13, 14, 19, 79, 89, 111, 125] [14, 20, 48, 59, 65, 80, 108, 112] [15, 16, 21, 32, 76, 80, 99, 118] [5, 16, 17, 22, 63, 77, 90, 100] [0, 6, 17, 23, 91, 101, 111, 114] [2, 16, 18, 19, 72, 83, 84, 94] [3, 17, 19, 20, 85, 95, 105, 116] [21, 22, 27, 38, 82, 86, 87, 97] [10, 22, 23, 28, 64, 83, 88, 98] [1, 11, 23, 29, 70, 89, 116, 119] [12, 25, 26, 31, 65, 86, 99, 109] [7, 25, 27, 28, 80, 92, 93, 103] [8, 26, 28, 29, 63, 67, 94, 104]

Code ID 126-26-9 · download JSON · raw on GitHub