[[198,22,7]] · QEC Challenge

[[198,22,7]] d ≤

198

kd²/n

5.444

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[17, 58, 59, 60, 61, 114, 115]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[56, 57, 61, 62, 114, 154, 159]

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 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×99 (2,2)×99 (3,2)×198 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 99 (1,4): 99 (2,2): 99 (2,4): 792 (2,6): 594 (3,2): 198 (3,4): 2970 (3,6): 9504 (3,8): 4455 (3,10): 396

trapping sets H_Z (1,2)×99 (2,2)×99 (3,2)×198 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 99 (1,4): 99 (2,2): 99 (2,4): 792 (2,6): 594 (3,2): 198 (3,4): 2970 (3,6): 9504 (3,8): 4455 (3,10): 396

Circuit tier

syndrome-extraction memory circuits committed under circuits/198-22-7/ · canonical noise recipe, 7 rounds, stim 1.16.0

d_circ ≤ 6 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 6 · fault-set witness of 6 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 6)

[51218, 51254, 51267, 51289, 51399, 51430]

d_circ^Z 6 · fault-set witness of 6 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 6)

[4159, 4161, 7219, 7222, 12139, 12173]

no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA coset code on D11xC9 (order 198) / H=[0, 63]: a = [0, 8, 117, 121], b in N_G(H) = [0, 65]

model Claude Opus 5.5 (claimed, not verified)

date 2026-10-17

family 2BGA coset (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

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

[[198,22,7]] — 2BGA coset code of index 99, check weight 6

Direction & hypothesis

Target: the unrestricted x weight-6 cell at n in 111..220 with check weight 6. Only board entries with check weight <= 6 can dominate a weight-6 code; at the time of the search no such entry had n' <= 198, k' >= 22 and d' >= 7, so d = 7 was enough here. The gate's novelty check reports that it advances the frontier (k); board-relative novelty from verify/validate_candidate.py, literature novelty unverified.

Hypothesis: 2BGA coset codes reach high k at small n because the right support only needs to lie in the normalizer of H; a weight-2 right support b = 1 + h with h in N_G(H) acts on the coset space G/H as a permutation, so its kernel (functions constant on the orbits of h) is large, and a weight-4 left support can share many of those zeros at check weight 6.

What was searched

2BGA coset codes with (|a|, |b|) = (2, 4) (100 random pairs per (G, H)) and (4, 2) (40 per (G, H)), b inside the normalizer, over direct products D_m x C_k, split metacyclic groups, A4/S4/A5/S5 x C_k and D_a x D_b with a cyclic non-normal subgroup H of index 56..110 and |N_G(H)| >= 6, check weight 6. Funnel for this job: 960579 candidates built, 84290 connected with k >= 8 screened at 400 gf2_fast RIS trials, 1 promoted to 20,000 trials, 1 survived it, 1 survived 200,000 trials and 1 survived 2,000,000 trials with the reading still at or above the needed d (a reading below the needed d is final, since the search returns an upper bound).

Evidence trail

Witness-backed upper bound d <= 7: the lightest logical operators found are embedded in the submission (distance.X / distance.Z witnesses) and are checked by the verifier. Confirmation ladder on the box (gf2_fast, research/kit/surrogate.distance_rand_witness, 20 threads):

| trials per side | lightest logical found | |---|---| | 400 / 20,000 / 200,000 / 2,000,000 funnel readings | 7 | | 4,000,000 fast RIS trials per side, fresh seed (63568) | 7 (X) | | 4,000,000 fast RIS trials per side, fresh seed (71487) | 7 (X) | | 4,000,000 fast RIS trials per side, fresh seed (79406) | 7 (X) |

Ladder verdict: corroborated (minimum over fresh seeds = 7, claimed 7). The claim is an upper bound; no exactness is asserted.

Dead ends

Weight-5 coset codes ((2, 3) and (3, 2) supports, same (G, H) menu): 1,232,557 built, only 14,534

with k >= 8; best readings tie the board ([[192,8,8]], [[168,8,7]]) or fall one short ([[216,10,6]], [[144,8,6]]).

Weight-7 coset codes ((2, 5), (5, 2), (3, 4), (4, 3)), about 1.23 million built: none reached the

weight-7 frontier at 400 trials.

Weight-6 two-block group-algebra codes with a = 1 + g on non-abelian groups of order 56..110

(126,495 built, 14,720 with k >= 8): best [[144,14,6]] and [[220,20,7]], two short of the frontier.

In this coset run (960,579 built, 84,290 connected with k >= 8) only one code survived the funnel;

the (2, 4) orientation produced nothing above the frontier.

Tools

Model: Claude Opus 5.5 (Claude Code agent, campaign agent A04). Brev n2d-highcpu-64 box, gf2_fast RIS (make fast) with 58 threads; research/kit modules nonabelian_lp / group_algebra / search / surrogate / submit / prescreen; gate = verify/validate_candidate.py via research/kit/coordination.gate_and_record. Approximate compute: 12 box minutes for the sweep plus the ladder above.

Reproduction

The code is the 2BGA coset code (research/kit/coset.py) on G = direct_product(dihedral(11)[0], cyclic_product(9)[0])[0] (order 198) modulo the non-normal cyclic subgroup H = [0, 63] ([G:H] = 99); a = [0, 8, 117, 121] acts on the left, b = [0, 65] (inside N_G(H)) on the right.

import sys; sys.path[:0] = ['research/kit', 'verify']
from group_algebra import dihedral, cyclic_product, direct_product, metacyclic, alt, sym
from coset import build_coset
mul = direct_product(dihedral(11)[0], cyclic_product(9)[0])[0]
HX, HZ = build_coset(mul, [0, 63], [0, 8, 117, 121], [0, 65], check_normalizer=False)

Witness search: research/kit/surrogate.lightest_logical (1,500 NumPy trials) tightened by gf2_fast RIS at the funnel depth; the ladder seeds are listed in the table above.

Parity checks

X-checks 99 (max weight 6) · Z-checks 99 (max weight 6)

H_X (99 checks, sparse supports)

[0, 1, 27, 32, 99, 106] [1, 2, 28, 33, 100, 107] [2, 3, 29, 34, 99, 101] [3, 4, 30, 35, 100, 102] [4, 5, 27, 31, 101, 103] [5, 6, 28, 32, 102, 104] [6, 7, 29, 33, 103, 105] [7, 8, 30, 34, 104, 106] [0, 8, 31, 35, 105, 107] [9, 10, 54, 59, 108, 115] [10, 11, 55, 60, 109, 116] [11, 12, 56, 61, 108, 110] [12, 13, 57, 62, 109, 111] [13, 14, 54, 58, 110, 112] [14, 15, 55, 59, 111, 113] [15, 16, 56, 60, 112, 114] [16, 17, 57, 61, 113, 115] [9, 17, 58, 62, 114, 116] [0, 5, 18, 19, 117, 124] [1, 6, 19, 20, 118, 125] [2, 7, 20, 21, 117, 119] [3, 8, 21, 22, 118, 120] [0, 4, 22, 23, 119, 121] [1, 5, 23, 24, 120, 122] [2, 6, 24, 25, 121, 123] [3, 7, 25, 26, 122, 124] [4, 8, 18, 26, 123, 125] [27, 28, 72, 77, 126, 133] [28, 29, 73, 78, 127, 134] [29, 30, 74, 79, 126, 128] [30, 31, 75, 80, 127, 129] [31, 32, 72, 76, 128, 130] [32, 33, 73, 77, 129, 131] [33, 34, 74, 78, 130, 132] [34, 35, 75, 79, 131, 133] [27, 35, 76, 80, 132, 134] [9, 14, 36, 37, 135, 142] [10, 15, 37, 38, 136, 143] [11, 16, 38, 39, 135, 137] [12, 17, 39, 40, 136, 138] [9, 13, 40, 41, 137, 139] [10, 14, 41, 42, 138, 140] [11, 15, 42, 43, 139, 141] [12, 16, 43, 44, 140, 142] [13, 17, 36, 44, 141, 143] [36, 41, 45, 46, 144, 151] [37, 42, 46, 47, 145, 152] [38, 43, 47, 48, 144, 146] [39, 44, 48, 49, 145, 147] [36, 40, 49, 50, 146, 148] [37, 41, 50, 51, 147, 149] [38, 42, 51, 52, 148, 150] [39, 43, 52, 53, 149, 151] [40, 44, 45, 53, 150, 152] [54, 55, 90, 95, 153, 160] [55, 56, 91, 96, 154, 161] [56, 57, 92, 97, 153, 155] [57, 58, 93, 98, 154, 156] [58, 59, 90, 94, 155, 157] [59, 60, 91, 95, 156, 158] [60, 61, 92, 96, 157, 159] [61, 62, 93, 97, 158, 160] [54, 62, 94, 98, 159, 161] [18, 23, 63, 64, 162, 169] [19, 24, 64, 65, 163, 170] [20, 25, 65, 66, 162, 164] [21, 26, 66, 67, 163, 165] [18, 22, 67, 68, 164, 166] [19, 23, 68, 69, 165, 167] [20, 24, 69, 70, 166, 168] [21, 25, 70, 71, 167, 169] [22, 26, 63, 71, 168, 170] [72, 73, 81, 86, 171, 178] [73, 74, 82, 87, 172, 179] [74, 75, 83, 88, 171, 173] [75, 76, 84, 89, 172, 174] [76, 77, 81, 85, 173, 175] [77, 78, 82, 86, 174, 176] [78, 79, 83, 87, 175, 177] [79, 80, 84, 88, 176, 178] [72, 80, 85, 89, 177, 179] [45, 50, 81, 82, 180, 187] [46, 51, 82, 83, 181, 188] [47, 52, 83, 84, 180, 182] [48, 53, 84, 85, 181, 183] [45, 49, 85, 86, 182, 184] [46, 50, 86, 87, 183, 185] [47, 51, 87, 88, 184, 186] [48, 52, 88, 89, 185, 187] [49, 53, 81, 89, 186, 188] [63, 68, 90, 91, 189, 196] [64, 69, 91, 92, 190, 197] [65, 70, 92, 93, 189, 191] [66, 71, 93, 94, 190, 192] [63, 67, 94, 95, 191, 193] [64, 68, 95, 96, 192, 194] [65, 69, 96, 97, 193, 195] [66, 70, 97, 98, 194, 196] [67, 71, 90, 98, 195, 197]

H_Z (99 checks, sparse supports)

[0, 2, 99, 107, 117, 121] [1, 3, 99, 100, 118, 122] [2, 4, 100, 101, 119, 123] [3, 5, 101, 102, 120, 124] [4, 6, 102, 103, 121, 125] [5, 7, 103, 104, 117, 122] [6, 8, 104, 105, 118, 123] [0, 7, 105, 106, 119, 124] [1, 8, 106, 107, 120, 125] [9, 11, 108, 116, 135, 139] [10, 12, 108, 109, 136, 140] [11, 13, 109, 110, 137, 141] [12, 14, 110, 111, 138, 142] [13, 15, 111, 112, 139, 143] [14, 16, 112, 113, 135, 140] [15, 17, 113, 114, 136, 141] [9, 16, 114, 115, 137, 142] [10, 17, 115, 116, 138, 143] [18, 20, 117, 125, 162, 166] [19, 21, 117, 118, 163, 167] [20, 22, 118, 119, 164, 168] [21, 23, 119, 120, 165, 169] [22, 24, 120, 121, 166, 170] [23, 25, 121, 122, 162, 167] [24, 26, 122, 123, 163, 168] [18, 25, 123, 124, 164, 169] [19, 26, 124, 125, 165, 170] [27, 29, 99, 103, 126, 134] [28, 30, 100, 104, 126, 127] [29, 31, 101, 105, 127, 128] [30, 32, 102, 106, 128, 129] [31, 33, 103, 107, 129, 130] [32, 34, 99, 104, 130, 131] [33, 35, 100, 105, 131, 132] [27, 34, 101, 106, 132, 133] [28, 35, 102, 107, 133, 134] [36, 38, 135, 143, 144, 148] [37, 39, 135, 136, 145, 149] [38, 40, 136, 137, 146, 150] [39, 41, 137, 138, 147, 151] [40, 42, 138, 139, 148, 152] [41, 43, 139, 140, 144, 149] [42, 44, 140, 141, 145, 150] [36, 43, 141, 142, 146, 151] [37, 44, 142, 143, 147, 152] [45, 47, 144, 152, 180, 184] [46, 48, 144, 145, 181, 185] [47, 49, 145, 146, 182, 186] [48, 50, 146, 147, 183, 187] [49, 51, 147, 148, 184, 188] [50, 52, 148, 149, 180, 185] [51, 53, 149, 150, 181, 186] [45, 52, 150, 151, 182, 187] [46, 53, 151, 152, 183, 188] [54, 56, 108, 112, 153, 161] [55, 57, 109, 113, 153, 154] [56, 58, 110, 114, 154, 155] [57, 59, 111, 115, 155, 156] [58, 60, 112, 116, 156, 157] [59, 61, 108, 113, 157, 158] [60, 62, 109, 114, 158, 159] [54, 61, 110, 115, 159, 160] [55, 62, 111, 116, 160, 161] [63, 65, 162, 170, 189, 193] [64, 66, 162, 163, 190, 194] [65, 67, 163, 164, 191, 195] [66, 68, 164, 165, 192, 196] [67, 69, 165, 166, 193, 197] [68, 70, 166, 167, 189, 194] [69, 71, 167, 168, 190, 195] [63, 70, 168, 169, 191, 196] [64, 71, 169, 170, 192, 197] [72, 74, 126, 130, 171, 179] [73, 75, 127, 131, 171, 172] [74, 76, 128, 132, 172, 173] [75, 77, 129, 133, 173, 174] [76, 78, 130, 134, 174, 175] [77, 79, 126, 131, 175, 176] [78, 80, 127, 132, 176, 177] [72, 79, 128, 133, 177, 178] [73, 80, 129, 134, 178, 179] [81, 83, 171, 175, 180, 188] [82, 84, 172, 176, 180, 181] [83, 85, 173, 177, 181, 182] [84, 86, 174, 178, 182, 183] [85, 87, 175, 179, 183, 184] [86, 88, 171, 176, 184, 185] [87, 89, 172, 177, 185, 186] [81, 88, 173, 178, 186, 187] [82, 89, 174, 179, 187, 188] [90, 92, 153, 157, 189, 197] [91, 93, 154, 158, 189, 190] [92, 94, 155, 159, 190, 191] [93, 95, 156, 160, 191, 192] [94, 96, 157, 161, 192, 193] [95, 97, 153, 158, 193, 194] [96, 98, 154, 159, 194, 195] [90, 97, 155, 160, 195, 196] [91, 98, 156, 161, 196, 197]

Code ID 198-22-7 · download JSON · raw on GitHub