[[240,32,8]] · QEC Challenge

[[240,32,8]] d ≤

240

kd²/n

8.533

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (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, 8 qubits)

[26, 72, 73, 111, 133, 182, 216, 221]

d_Z 8 · witness weight 8 (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, 8 qubits)

[123, 133, 134, 153, 174, 176, 198, 199]

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 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)×120 (2,4)×360 (3,3)×120 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 120 (1,4): 120 (2,4): 360 (2,5): 1440 (2,6): 720 (3,3): 120 (3,5): 1320 (3,6): 12720 (3,7): 18120 (3,8): 6480 (3,9): 2160 (3,10): 480

trapping sets H_Z (1,3)×120 (2,4)×360 (3,3)×120 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 120 (1,4): 120 (2,4): 360 (2,5): 1440 (2,6): 720 (3,3): 120 (3,5): 1320 (3,6): 12720 (3,7): 18120 (3,8): 6480 (3,9): 2160 (3,10): 480

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C30 x C2 x C2 = SmallGroup(120, 47) (order 120), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 240; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 25, 26, 112}, b = {0, 16, 82} 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 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

[[240,32,8]] — two-block group-algebra code over C30 x C2 x C2

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| = 240

Parameters are also spelled out in provenance.construction of codes/240-32-8.json.

Parity checks

X-checks 120 (max weight 7) · Z-checks 120 (max weight 7)

H_X (120 checks, sparse supports)

[0, 29, 47, 48, 120, 154, 221] [1, 36, 40, 60, 121, 165, 229] [2, 16, 31, 65, 122, 171, 232] [3, 28, 49, 68, 123, 174, 233] [4, 10, 51, 70, 124, 125, 179] [5, 52, 71, 72, 125, 135, 177] [6, 24, 42, 53, 126, 183, 236] [7, 39, 56, 61, 127, 186, 237] [8, 18, 58, 63, 128, 129, 191] [9, 59, 64, 83, 129, 143, 189] [10, 15, 32, 85, 130, 192, 238] [3, 11, 34, 87, 131, 132, 197] [12, 35, 54, 88, 132, 148, 195] [2, 13, 72, 90, 133, 134, 200] [14, 51, 73, 91, 134, 151, 198] [12, 15, 26, 92, 135, 136, 202] [16, 27, 75, 93, 120, 136, 137] [1, 17, 76, 94, 137, 154, 201] [18, 23, 43, 74, 138, 203, 239] [7, 19, 45, 76, 139, 140, 208] [20, 46, 66, 77, 140, 159, 206] [6, 21, 79, 83, 141, 142, 211] [22, 63, 80, 84, 142, 162, 209] [20, 23, 37, 81, 143, 144, 213] [24, 38, 82, 86, 121, 144, 145] [0, 25, 87, 102, 145, 165, 212] [0, 26, 54, 103, 146, 147, 216] [27, 34, 55, 104, 147, 168, 214] [5, 13, 28, 105, 148, 149, 218] [14, 29, 57, 106, 122, 149, 150] [6, 30, 58, 78, 150, 171, 217] [11, 27, 31, 107, 151, 152, 220] [12, 32, 94, 108, 123, 152, 153] [7, 33, 75, 95, 153, 174, 219] [30, 34, 49, 109, 124, 154, 155] [8, 35, 50, 97, 125, 155, 156] [9, 36, 98, 110, 156, 177, 221] [1, 37, 66, 95, 157, 158, 224] [38, 45, 67, 96, 158, 180, 222] [9, 21, 39, 97, 159, 160, 226] [22, 40, 69, 98, 126, 160, 161] [2, 41, 70, 89, 161, 183, 225] [19, 38, 42, 99, 162, 163, 228] [20, 43, 100, 102, 127, 163, 164] [3, 44, 86, 103, 164, 186, 227] [41, 45, 61, 101, 128, 165, 166] [4, 46, 62, 105, 129, 166, 167] [5, 47, 106, 114, 167, 189, 229] [4, 14, 48, 114, 168, 169, 231] [5, 49, 78, 115, 130, 169, 170] [18, 50, 57, 79, 170, 192, 230] [17, 32, 51, 116, 131, 171, 172] [19, 33, 52, 81, 132, 172, 173] [20, 53, 82, 99, 173, 195, 232] [29, 50, 54, 117, 133, 174, 175] [21, 30, 55, 110, 134, 175, 176] [22, 56, 97, 111, 176, 198, 233] [23, 53, 57, 73, 136, 177, 178] [24, 58, 74, 112, 137, 178, 179] [11, 25, 59, 118, 135, 179, 201] [8, 22, 60, 110, 180, 181, 235] [9, 61, 89, 111, 138, 181, 182] [10, 62, 69, 90, 182, 203, 234] [25, 43, 63, 112, 139, 183, 184] [11, 44, 64, 92, 140, 184, 185] [12, 65, 93, 107, 185, 206, 236] [40, 62, 66, 113, 141, 186, 187] [13, 41, 67, 114, 142, 187, 188] [14, 68, 105, 115, 188, 209, 237] [15, 65, 69, 84, 144, 189, 190] [16, 70, 85, 116, 145, 190, 191] [17, 19, 71, 119, 143, 191, 212] [16, 33, 72, 119, 146, 192, 193] [17, 37, 73, 99, 147, 193, 194] [38, 74, 81, 100, 194, 214, 238] [36, 39, 55, 75, 149, 195, 196] [40, 56, 76, 101, 150, 196, 197] [4, 41, 77, 113, 148, 197, 217] [42, 52, 74, 78, 152, 198, 199] [43, 53, 79, 118, 153, 199, 200] [26, 44, 80, 112, 151, 200, 219] [45, 77, 81, 95, 155, 201, 202] [28, 46, 82, 96, 120, 156, 202] [24, 44, 83, 118, 157, 203, 204] [25, 26, 84, 107, 158, 204, 205] [27, 85, 92, 108, 205, 222, 239] [28, 47, 67, 86, 160, 206, 207] [29, 68, 87, 109, 161, 207, 208] [8, 30, 88, 117, 159, 208, 225] [31, 64, 85, 89, 163, 209, 210] [32, 65, 90, 119, 164, 210, 211] [33, 37, 91, 116, 162, 211, 227] [34, 88, 92, 103, 166, 212, 213] [35, 39, 93, 104, 121, 167, 213] [35, 56, 60, 94, 169, 214, 215] [36, 61, 95, 113, 170, 215, 216] [13, 62, 96, 101, 168, 216, 230] [59, 63, 79, 97, 172, 217, 218] [15, 64, 80, 98, 122, 173, 218] [66, 76, 96, 99, 175, 219, 220] [48, 67, 77, 100, 123, 176, 220] [2, 69, 101, 111, 124, 178, 221] [46, 48, 68, 102, 181, 222, 223] [47, 49, 103, 117, 182, 223, 224] [21, 50, 104, 109, 180, 224, 234] [51, 71, 90, 105, 184, 225, 226] [23, 52, 91, 106, 126, 185, 226] [54, 87, 104, 107, 187, 227, 228] [55, 60, 88, 108, 127, 188, 228] [6, 57, 109, 115, 128, 190, 229] [58, 80, 83, 110, 193, 230, 231] [31, 59, 84, 111, 130, 194, 231] [0, 86, 100, 112, 131, 196, 232] [10, 89, 98, 113, 133, 199, 233] [70, 72, 91, 114, 204, 234, 235] [42, 71, 73, 115, 138, 205, 235] [1, 75, 108, 116, 139, 207, 236] [18, 78, 106, 117, 141, 210, 237] [3, 82, 102, 118, 146, 215, 238] [7, 93, 94, 119, 157, 223, 239]

H_Z (120 checks, sparse supports)

[0, 16, 82, 120, 145, 146, 232] [1, 24, 93, 121, 137, 157, 236] [2, 29, 98, 122, 133, 161, 221] [3, 32, 100, 123, 131, 164, 238] [4, 34, 101, 124, 166, 168, 197] [4, 5, 35, 125, 148, 167, 169] [6, 40, 106, 126, 141, 150, 229] [7, 43, 108, 127, 139, 153, 239] [8, 45, 109, 128, 155, 180, 208] [8, 9, 46, 129, 156, 159, 181] [10, 49, 111, 124, 130, 182, 233] [11, 51, 112, 131, 151, 179, 184] [11, 12, 52, 132, 135, 152, 185] [13, 54, 113, 133, 148, 187, 216] [13, 14, 55, 134, 149, 168, 188] [5, 15, 59, 130, 135, 189, 218] [15, 16, 57, 122, 136, 190, 192] [16, 17, 58, 137, 171, 191, 193] [18, 61, 115, 128, 138, 170, 237] [19, 63, 116, 139, 162, 172, 191] [19, 20, 64, 140, 143, 163, 173] [21, 66, 117, 141, 159, 175, 224] [21, 22, 67, 142, 160, 176, 180] [9, 23, 71, 138, 143, 177, 226] [23, 24, 69, 126, 144, 178, 203] [24, 25, 70, 145, 179, 183, 204] [26, 72, 118, 135, 146, 200, 204] [26, 27, 73, 136, 147, 151, 205] [12, 28, 77, 123, 148, 202, 206] [28, 29, 75, 120, 149, 174, 207] [29, 30, 76, 150, 154, 175, 208] [14, 31, 80, 122, 151, 209, 231] [31, 32, 78, 130, 152, 171, 210] [32, 33, 79, 153, 172, 192, 211] [0, 17, 34, 131, 147, 154, 212] [34, 35, 81, 132, 155, 213, 214] [35, 36, 82, 121, 156, 195, 215] [37, 83, 119, 143, 157, 193, 211] [37, 38, 84, 144, 158, 162, 194] [20, 39, 88, 127, 159, 195, 213] [39, 40, 86, 121, 160, 186, 196] [40, 41, 87, 161, 165, 187, 197] [22, 42, 91, 126, 162, 198, 235] [42, 43, 89, 138, 163, 183, 199] [43, 44, 90, 164, 184, 200, 203] [1, 25, 45, 139, 158, 165, 201] [45, 46, 92, 140, 166, 202, 222] [46, 47, 93, 120, 167, 206, 223] [27, 48, 96, 120, 168, 220, 222] [48, 49, 94, 123, 154, 169, 223] [49, 50, 95, 155, 170, 174, 224] [2, 30, 51, 124, 134, 171, 225] [51, 52, 97, 125, 172, 198, 226] [52, 53, 98, 126, 173, 177, 199] [3, 33, 54, 132, 146, 174, 227] [54, 55, 99, 147, 175, 195, 228] [55, 56, 100, 127, 176, 196, 214] [5, 36, 57, 149, 170, 177, 229] [57, 58, 101, 128, 150, 178, 230] [4, 58, 59, 129, 179, 217, 231] [38, 60, 104, 121, 180, 214, 228] [60, 61, 102, 127, 165, 181, 215] [61, 62, 103, 166, 182, 186, 216] [6, 41, 63, 128, 142, 183, 217] [63, 64, 105, 129, 184, 209, 218] [64, 65, 106, 122, 185, 189, 210] [7, 44, 66, 140, 157, 186, 219] [66, 67, 107, 158, 187, 206, 220] [67, 68, 108, 123, 188, 207, 222] [9, 47, 69, 160, 182, 189, 221] [69, 70, 109, 124, 161, 190, 234] [8, 70, 71, 125, 191, 225, 235] [10, 50, 72, 125, 133, 192, 234] [72, 73, 110, 134, 177, 193, 235] [73, 74, 111, 138, 178, 194, 198] [12, 53, 75, 136, 153, 195, 236] [75, 76, 112, 137, 139, 196, 219] [11, 76, 77, 140, 197, 201, 220] [14, 56, 78, 150, 169, 198, 237] [78, 79, 113, 141, 170, 199, 217] [13, 79, 80, 142, 200, 218, 230] [17, 59, 81, 143, 172, 194, 201] [15, 81, 82, 144, 173, 202, 238] [18, 62, 83, 129, 141, 203, 230] [83, 84, 114, 142, 189, 204, 231] [84, 85, 115, 130, 190, 205, 209] [20, 65, 86, 144, 164, 206, 232] [86, 87, 116, 131, 145, 207, 227] [19, 87, 88, 132, 208, 212, 228] [22, 68, 89, 161, 181, 209, 233] [89, 90, 117, 133, 182, 210, 225] [21, 90, 91, 134, 211, 226, 234] [25, 71, 92, 135, 184, 205, 212] [23, 92, 93, 136, 185, 213, 239] [27, 74, 94, 137, 152, 214, 239] [94, 95, 118, 153, 157, 201, 215] [26, 95, 96, 158, 202, 216, 219] [30, 77, 97, 155, 159, 176, 217] [28, 97, 98, 156, 160, 218, 233] [33, 80, 99, 162, 173, 193, 219] [31, 99, 100, 163, 194, 220, 232] [0, 36, 101, 165, 196, 216, 221] [38, 85, 102, 145, 163, 222, 238] [102, 103, 119, 146, 164, 212, 223] [37, 103, 104, 147, 213, 224, 227] [41, 88, 105, 148, 166, 188, 225] [39, 105, 106, 149, 167, 226, 237] [44, 91, 107, 151, 185, 204, 227] [42, 107, 108, 152, 205, 228, 236] [1, 47, 109, 154, 207, 224, 229] [50, 96, 110, 156, 175, 180, 230] [48, 110, 111, 176, 181, 221, 231] [2, 53, 112, 178, 183, 200, 232] [3, 56, 113, 186, 197, 215, 233] [62, 104, 114, 167, 168, 187, 234] [60, 114, 115, 169, 188, 229, 235] [6, 65, 116, 171, 190, 211, 236] [7, 68, 117, 174, 208, 223, 237] [10, 74, 118, 179, 199, 203, 238] [18, 85, 119, 191, 192, 210, 239]

Code ID 240-32-8 · download JSON · raw on GitHub