[[216,10,18]] · QEC Challenge

[[216,10,18]] d ≤

216

kd²/n

15.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 18 · witness weight 18 (claimed upper_bound)

witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12

witness operator (support, 18 qubits)

[4, 5, 24, 35, 37, 39, 42, 44, 70, 98, 112, 122, 134, 145, 164, 176, 189, 196]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12

witness operator (support, 18 qubits)

[27, 61, 71, 76, 77, 81, 123, 130, 138, 141, 148, 157, 181, 192, 200, 207, 210, 215]

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×108 (2,3)×36 (3,4)×396 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 108 (1,4): 108 (2,3): 36 (2,4): 324 (2,5): 1224 (2,6): 648 (3,4): 396 (3,5): 1980 (3,6): 10152 (3,7): 15012 (3,8): 6336 (3,9): 1728 (3,10): 432

trapping sets H_Z (1,3)×108 (2,3)×36 (3,4)×396 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 108 (1,4): 108 (2,3): 36 (2,4): 324 (2,5): 1224 (2,6): 648 (3,4): 396 (3,5): 1980 (3,6): 10152 (3,7): 15012 (3,8): 6336 (3,9): 1728 (3,10): 432

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C18 x S3 = SmallGroup(108, 24) (order 108), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 216; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 8, 62, 80}, b = {0, 22, 57} as element indices. Check weight 7. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race.

model Mimo-V2.6-Flash (claimed, not verified)

reference arXiv:2610.13880

date 2026-10-12

notes Parameters and construction taken from the public database of arXiv:2610.13880v1. Distance is an upper bound certified by the witnesses attached here. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0; rebuilt here with the repo's own group data. Distance is an upper bound certified by the attached witnesses, not the paper's exact distance.

family 2bga-nonabelian (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

[[216,10,18]] — two-block group-algebra code over C18 x S3

Direction & hypothesis

Cell weight-8 x unrestricted, CSS board. The argument is coverage, not discovery: this code is published in arXiv:2610.13880v1 (Lee, "Bicycle Race"), and nothing on the board occupied this (n, k, d, w) point. It advances the four-axis Pareto frontier by moving into a cell that was empty at these parameters.

What was searched

Not a search for new codes. The arXiv:2610.13880v1 database (github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0, 12,448 records) was rebuilt here from its group and block data, then screened against every CSS entry on main under the site's Pareto rule (n <=, k >=, d >=, w <= with at least one strict) within the same weight class. Records whose parameters the paper already tabulates, and candidates claimed by other open filings, were removed first. Distances were re-searched locally:

X side: research/kit/surrogate.lightest_logical at 2,000 trials
Z side: research/kit/surrogate.lightest_logical at 2,000 trials

Evidence trail

Witness-backed upper bound, not an exact distance. The attached X and Z operators were found by the searches above and re-checked by the verifier's own criteria (in ker(opposite) and outside rowspace(own)). verify/validate_candidate.py reports passed: true and board_advancing: true, with d_only_gain: false.

The paper states an exact distance for this record, verified there with a second exact engine plus GAP rebuild and independent upper-bound attacks. That exactness is not claimed here; the board records what the witness certifies.

Dead ends

Direct sums of two board codes: rejected, CONTRIBUTING.md requires a single

connected Tanner graph and a connected stabilizer group.

Fitting a single-layer 2D layout to reach the local-2d cells: searched with

research/local2d/fold_layout.search_layout, best radii 10–13 against a ceiling of 4.0, so these codes stay unrestricted.

Hypergraph products of two parents from this pool: connectivity passes, but n

multiplies (two ~250-qubit parents give n = 80,010).

Tools

Model: Mimo-V2.6-Flash (self-reported, provenance.model), driving OpenCode. Rebuild: research/kit/group_algebra.build_2bga from the database's data/groups.jsonl Cayley table. Witness search: research/kit/surrogate (numpy backend, 2,000 trials/side screening) with the gf2_fast accelerator for longer runs. Gate: research/kit/coordination.py gate, which runs verify/validate_candidate.py unchanged. Search run by @mathysrennela.

Reproduction

from group_algebra import build_2bga
# a and b are element indices from github.com/gideonlsx/bicycle-race @ aed705a data/codes.jsonl
HX, HZ = build_2bga(mul, a, b)   # mul from data/groups.jsonl, n = 2|G| = 216

Parameters are also spelled out in provenance.construction of codes/216-10-18.json.

Parity checks

X-checks 108 (max weight 7) · Z-checks 108 (max weight 7)

H_X (108 checks, sparse supports)

[0, 10, 25, 65, 108, 157, 207] [1, 16, 19, 50, 109, 144, 211] [2, 3, 44, 84, 110, 177, 195] [3, 28, 49, 87, 111, 182, 193] [1, 4, 29, 88, 112, 115, 214] [5, 14, 30, 52, 113, 162, 205] [6, 7, 31, 70, 114, 164, 204] [7, 36, 41, 75, 115, 169, 202] [0, 8, 42, 77, 111, 116, 215] [9, 39, 43, 78, 117, 175, 196] [10, 13, 69, 99, 118, 174, 197] [6, 11, 14, 100, 119, 127, 209] [12, 15, 29, 72, 120, 142, 212] [11, 13, 66, 74, 109, 121, 208] [7, 14, 54, 101, 122, 130, 141] [15, 34, 55, 77, 123, 139, 213] [8, 16, 38, 56, 124, 131, 190] [9, 17, 36, 57, 125, 159, 188] [18, 37, 58, 79, 126, 185, 189] [19, 22, 56, 90, 127, 187, 188] [2, 20, 23, 92, 118, 128, 213] [21, 24, 59, 93, 129, 155, 208] [20, 22, 52, 61, 108, 130, 212] [3, 23, 67, 95, 121, 131, 154] [24, 64, 68, 96, 132, 152, 209] [4, 24, 25, 69, 122, 133, 199] [5, 26, 70, 97, 134, 146, 197] [23, 27, 66, 71, 135, 172, 198] [4, 28, 85, 89, 114, 136, 196] [19, 29, 34, 106, 126, 137, 149] [30, 35, 54, 92, 124, 138, 215] [20, 31, 36, 58, 139, 150, 171] [21, 32, 37, 56, 140, 169, 179] [33, 38, 57, 94, 141, 170, 200] [18, 22, 31, 34, 116, 142, 166] [16, 32, 35, 95, 135, 143, 164] [23, 36, 63, 80, 144, 155, 168] [24, 37, 61, 81, 110, 145, 184] [38, 62, 82, 97, 146, 167, 203] [3, 26, 39, 83, 136, 147, 186] [27, 40, 64, 84, 132, 148, 206] [8, 41, 72, 80, 110, 149, 205] [10, 42, 47, 102, 135, 136, 150] [43, 48, 83, 103, 133, 151, 214] [11, 43, 44, 49, 137, 152, 184] [12, 45, 50, 104, 153, 166, 182] [42, 46, 51, 85, 154, 183, 191] [9, 13, 44, 47, 112, 155, 179] [45, 48, 79, 86, 126, 156, 177] [14, 48, 49, 89, 142, 157, 181] [15, 50, 90, 105, 114, 158, 171] [47, 51, 88, 91, 159, 180, 194] [17, 51, 52, 92, 149, 160, 173] [18, 49, 53, 93, 123, 161, 210] [16, 33, 41, 54, 128, 146, 162] [17, 31, 55, 102, 144, 154, 163] [42, 56, 61, 82, 148, 164, 175] [43, 57, 62, 80, 108, 165, 199] [58, 63, 81, 104, 147, 166, 210] [10, 45, 59, 64, 121, 167, 201] [46, 60, 65, 83, 151, 168, 194] [2, 40, 47, 61, 133, 169, 192] [0, 48, 59, 62, 118, 161, 170] [39, 60, 63, 105, 117, 171, 191] [13, 50, 64, 98, 119, 172, 204] [51, 65, 86, 99, 120, 156, 173] [15, 53, 66, 100, 158, 163, 174] [21, 25, 28, 67, 119, 159, 175] [26, 68, 94, 98, 141, 157, 176] [29, 68, 69, 74, 161, 162, 177] [30, 70, 75, 107, 109, 178, 190] [67, 71, 76, 100, 160, 179, 206] [32, 71, 72, 77, 130, 180, 192] [33, 69, 73, 78, 138, 181, 203] [6, 26, 34, 74, 124, 182, 201] [27, 35, 72, 75, 127, 148, 183] [25, 73, 76, 101, 113, 184, 200] [37, 76, 77, 102, 128, 185, 195] [38, 74, 78, 103, 129, 143, 186] [7, 40, 79, 104, 145, 176, 187] [0, 60, 67, 80, 152, 173, 188] [2, 39, 68, 81, 111, 181, 189] [40, 59, 82, 107, 129, 172, 190] [28, 70, 83, 86, 112, 191, 211] [71, 84, 87, 98, 113, 176, 192] [30, 73, 85, 88, 143, 178, 193] [4, 12, 75, 86, 137, 187, 194] [5, 76, 85, 87, 134, 138, 195] [35, 78, 88, 106, 140, 183, 196] [1, 45, 54, 89, 139, 186, 197] [46, 52, 55, 90, 115, 168, 198] [44, 53, 91, 106, 120, 185, 199] [57, 91, 92, 95, 116, 200, 207] [58, 89, 93, 96, 117, 163, 201] [19, 60, 94, 97, 156, 165, 202] [21, 53, 62, 95, 150, 174, 203] [1, 63, 94, 96, 125, 151, 204] [22, 65, 97, 107, 153, 170, 205] [5, 11, 90, 98, 122, 202, 206] [12, 66, 91, 99, 123, 153, 207] [55, 93, 100, 101, 125, 198, 208] [17, 33, 96, 101, 160, 165, 209] [9, 73, 81, 102, 131, 193, 210] [6, 79, 82, 103, 132, 140, 211] [41, 84, 104, 105, 134, 189, 212] [8, 46, 87, 105, 147, 178, 213] [18, 32, 103, 106, 145, 180, 214] [20, 27, 99, 107, 158, 167, 215]

H_Z (108 checks, sparse supports)

[0, 22, 57, 108, 116, 170, 188] [1, 13, 70, 109, 112, 197, 204] [2, 37, 41, 110, 128, 169, 189] [3, 8, 81, 110, 111, 131, 147] [4, 47, 83, 112, 133, 136, 194] [5, 76, 84, 113, 134, 195, 206] [6, 28, 50, 114, 119, 182, 211] [4, 7, 90, 114, 115, 122, 187] [8, 34, 92, 116, 124, 149, 213] [9, 63, 93, 117, 125, 155, 210] [10, 20, 62, 108, 118, 150, 167] [11, 64, 67, 119, 121, 152, 206] [12, 65, 91, 120, 153, 194, 207] [13, 23, 59, 118, 121, 155, 172] [14, 25, 98, 113, 119, 122, 157] [15, 53, 99, 120, 123, 158, 174] [16, 30, 74, 109, 124, 143, 162] [17, 96, 100, 125, 160, 163, 209] [18, 29, 48, 126, 142, 161, 214] [11, 19, 75, 109, 127, 137, 202] [20, 54, 77, 128, 130, 139, 215] [21, 78, 82, 129, 140, 175, 203] [14, 22, 72, 127, 130, 142, 205] [16, 23, 102, 128, 131, 135, 144] [24, 40, 103, 129, 132, 133, 145] [25, 43, 61, 108, 133, 175, 184] [26, 87, 104, 134, 147, 176, 182] [27, 35, 42, 135, 148, 183, 215] [28, 39, 42, 111, 136, 175, 191] [29, 44, 86, 112, 120, 137, 177] [30, 73, 87, 113, 138, 178, 193] [15, 31, 89, 114, 139, 142, 163] [32, 88, 103, 140, 143, 180, 214] [14, 33, 68, 141, 162, 181, 209] [12, 34, 49, 123, 137, 142, 182] [35, 78, 85, 138, 143, 183, 196] [1, 36, 55, 115, 125, 139, 144] [37, 79, 106, 126, 140, 145, 185] [26, 38, 54, 124, 141, 146, 186] [39, 58, 105, 117, 147, 171, 189] [40, 56, 75, 148, 169, 187, 190] [29, 41, 52, 115, 149, 162, 212] [31, 42, 95, 116, 150, 154, 164] [43, 60, 96, 117, 151, 152, 165] [24, 44, 80, 110, 152, 155, 199] [45, 97, 99, 153, 156, 167, 197] [23, 46, 55, 154, 168, 198, 213] [21, 36, 47, 150, 155, 159, 169] [48, 65, 94, 151, 156, 157, 170] [0, 49, 68, 111, 152, 157, 161] [50, 66, 107, 109, 153, 158, 172] [17, 51, 67, 154, 159, 160, 173] [52, 71, 101, 113, 130, 160, 198] [53, 62, 69, 161, 174, 199, 203] [5, 54, 69, 122, 138, 162, 197] [55, 66, 93, 123, 163, 198, 208] [6, 35, 56, 124, 127, 140, 164] [57, 94, 101, 125, 141, 165, 200] [34, 45, 58, 126, 139, 166, 201] [38, 59, 107, 129, 167, 170, 190] [36, 60, 90, 168, 171, 188, 202] [7, 32, 61, 130, 145, 164, 169] [33, 62, 97, 146, 165, 170, 203] [31, 50, 63, 144, 166, 171, 204] [27, 64, 82, 132, 148, 167, 172] [52, 65, 80, 108, 168, 173, 205] [10, 66, 95, 121, 135, 174, 207] [9, 56, 67, 131, 175, 179, 188] [68, 79, 84, 132, 176, 177, 189] [2, 48, 69, 118, 133, 177, 181] [70, 85, 105, 114, 134, 178, 191] [32, 47, 71, 135, 179, 180, 192] [51, 72, 106, 120, 149, 180, 183] [49, 73, 81, 181, 184, 193, 210] [3, 45, 74, 121, 177, 182, 186] [46, 75, 88, 115, 178, 183, 194] [37, 44, 76, 179, 184, 185, 195] [18, 77, 91, 116, 123, 180, 185] [39, 78, 89, 117, 181, 186, 196] [19, 79, 86, 126, 156, 187, 211] [17, 19, 80, 144, 149, 165, 188] [18, 81, 104, 145, 166, 189, 210] [16, 70, 82, 146, 164, 190, 211] [46, 63, 83, 147, 151, 168, 191] [61, 72, 84, 110, 148, 192, 212] [3, 85, 102, 136, 154, 193, 195] [51, 60, 86, 156, 173, 191, 194] [2, 77, 87, 111, 192, 195, 213] [9, 28, 88, 112, 159, 193, 196] [10, 26, 89, 136, 157, 197, 201] [27, 90, 100, 127, 158, 198, 206] [25, 57, 91, 159, 199, 200, 207] [33, 76, 92, 128, 138, 160, 200] [59, 74, 93, 129, 161, 201, 208] [7, 94, 98, 141, 176, 202, 204] [38, 73, 95, 131, 143, 200, 203] [6, 64, 96, 132, 201, 204, 209] [5, 41, 97, 134, 146, 202, 205] [40, 71, 98, 172, 176, 192, 206] [0, 92, 99, 118, 173, 207, 215] [13, 21, 100, 119, 174, 179, 208] [11, 24, 101, 122, 184, 208, 209] [53, 58, 102, 150, 163, 185, 210] [1, 83, 103, 151, 186, 211, 214] [12, 22, 104, 153, 166, 187, 212] [15, 20, 105, 158, 171, 212, 213] [4, 43, 106, 137, 196, 199, 214] [8, 30, 107, 178, 190, 205, 215]

Code ID 216-10-18 · download JSON · raw on GitHub