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[[126,18,14]] d ≤
n
126
k
18
d
14
kd²/n
28.0
w
10

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Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[0, 4, 8, 15, 22, 28, 51, 81, 90, 92, 102, 108, 110, 115]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[7, 18, 26, 31, 39, 40, 45, 56, 76, 77, 99, 107, 118, 123]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @willzeng
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle (2BGA on Z_63), circulant size 63, a(x) = 1 + x14 + x16 + x21 + x27, b(x) = 1 + x31 + x39 + x40 + x45. Weight-5 supports designed as multiples of a degree-9 divisor g of x^N-1 (k = 2 deg g). Check weight 10 (weight-9plus cell).
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:1904.02703, arXiv:2306.16400
date 2026-07-14
notes Support-5 2BGA campaign 20260714 r1-w9plus. Screen 1.5k trials; ladder [8000, 60000, 200000, 1000000] -> d [14, 14, 14, 14]; claimed d = lightest logical witnessed at 1000000 trials/side (gf2_fast RIS, seed 424242). Independent adversarial re-check: 3 fresh seeds (900001-900003) x 1M trials/side, lightest logical 14 each time, nothing lighter. Upper bound only. Lit-check 2026-07-14: no published [[126,18,14]] w<=10 GB found; closest [[126,28,8]] (w8, arXiv:2203.17216) and [[126,12,10]]/[[126,14,10]]; novelty vs literature unverified.
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

[[126,18,14]] — designed-divisor weight-10 GB on Z_63

Direction & hypothesis

First find of the designed-divisor track (see notes/258-32-22.md for the full campaign): weight-9plus cell, k pinned by a degree-9 divisor g | x^63−1 (k = 18), weight-5 multiples of g as supports.

What was searched

Designed-divisor sweep at N=63 within the 13,200-candidate campaign; screen at 1.5k RIS trials. This code: a(x) = 1+x¹⁴+x¹⁶+x²¹+x²⁷, b(x) = 1+x³¹+x³⁹+x⁴⁰+x⁴⁵.

Evidence trail

Ladder 8k → 60k → 200k → 1M trials/side: lightest logical 14 at every rung (seed 424242); claimed d = witnessed weight-14 logicals. Pre-submission adversarial re-check: 3 fresh seeds × 1M trials/side, all 14. Claim: upper bound d ≤ 14. At n=126 the gate's refutation search is meaningful, and the weekly board sweep re-tests with fresh seeds.

Dead ends

n=126 sits inside Lin–Pryadko's exhaustively enumerated range for W ≤ 8, so the lit check mattered: no published [[126,18,14]] at w ≤ 10 was found (closest: [[126,28,8]] at w8, [[126,12,10]]/[[126,14,10]]); weight-10 is outside the exhaustive enumerations. Novelty vs literature: unverified.

Tools

Claude Fable 5 agent campaign, research/ kit, gf2_fast.

Reproduction

GB on Z_63 with the supports above; construction string in codes/126-18-14.json.

Parity checks

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