[[252,30,10]] · QEC Challenge

[[252,30,10]] d ≤

252

kd²/n

11.905

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (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, 10 qubits)

[19, 52, 53, 70, 86, 88, 102, 117, 122, 123]

d_Z 10 · witness weight 10 (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, 10 qubits)

[128, 129, 146, 147, 149, 162, 183, 199, 201, 216]

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)×126 (2,4)×378 (3,3)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 126 (1,4): 126 (2,4): 378 (2,5): 1512 (2,6): 756 (3,3): 126 (3,5): 1512 (3,6): 12978 (3,7): 18648 (3,8): 7938 (3,9): 2268 (3,10): 504

trapping sets H_Z (1,3)×126 (2,4)×378 (3,3)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 126 (1,4): 126 (2,4): 378 (2,5): 1512 (2,6): 756 (3,3): 126 (3,5): 1512 (3,6): 12978 (3,7): 18648 (3,8): 7938 (3,9): 2268 (3,10): 504

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C7 x ((C3 x C3) : C2) = SmallGroup(126, 14) (order 126), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 252; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 5, 56, 119}, b = {0, 9, 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

[[252,30,10]] — two-block group-algebra code over C7 x ((C3 x C3) : 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| = 252

Parameters are also spelled out in provenance.construction of codes/252-30-10.json.

Parity checks

X-checks 126 (max weight 7) · Z-checks 126 (max weight 7)

H_X (126 checks, sparse supports)

[0, 42, 59, 81, 126, 181, 232] [1, 47, 52, 72, 127, 190, 238] [1, 2, 60, 77, 128, 137, 199] [3, 10, 79, 98, 129, 198, 201] [4, 20, 62, 99, 130, 202, 244] [0, 5, 65, 70, 131, 143, 208] [6, 67, 71, 90, 132, 207, 210] [7, 15, 68, 91, 133, 211, 247] [5, 8, 78, 94, 134, 149, 216] [6, 9, 22, 96, 126, 135, 218] [7, 10, 36, 80, 136, 152, 219] [11, 24, 97, 112, 137, 162, 216] [12, 25, 37, 113, 138, 217, 221] [13, 23, 38, 114, 139, 222, 250] [2, 14, 83, 88, 140, 157, 224] [3, 15, 85, 89, 127, 141, 226] [4, 16, 29, 86, 142, 160, 227] [17, 33, 87, 106, 143, 171, 224] [18, 31, 88, 107, 144, 225, 229] [19, 28, 32, 108, 145, 230, 251] [14, 20, 95, 108, 129, 146, 165] [15, 21, 38, 110, 128, 147, 232] [16, 22, 54, 97, 148, 168, 233] [17, 23, 40, 111, 129, 149, 180] [18, 24, 41, 55, 130, 150, 235] [19, 25, 39, 56, 151, 170, 236] [26, 43, 57, 121, 152, 182, 233] [2, 27, 58, 122, 153, 234, 237] [8, 28, 99, 104, 132, 154, 174] [9, 29, 101, 105, 131, 155, 238] [10, 30, 46, 102, 156, 177, 239] [11, 31, 50, 103, 132, 157, 189] [12, 32, 48, 104, 133, 158, 241] [13, 33, 45, 49, 159, 179, 242] [5, 34, 105, 118, 160, 191, 239] [35, 46, 51, 119, 161, 240, 243] [13, 28, 36, 109, 135, 162, 183] [29, 37, 56, 119, 134, 137, 163] [30, 38, 72, 111, 138, 164, 186] [31, 39, 58, 120, 135, 165, 198] [32, 40, 59, 73, 136, 166, 244] [33, 41, 57, 74, 167, 188, 245] [34, 42, 61, 75, 138, 168, 200] [8, 35, 43, 76, 139, 169, 246] [9, 44, 78, 125, 170, 203, 245] [7, 20, 45, 116, 141, 171, 192] [21, 46, 113, 117, 140, 143, 172] [22, 47, 64, 114, 144, 173, 195] [23, 48, 68, 115, 141, 174, 207] [24, 49, 66, 116, 142, 175, 247] [25, 50, 63, 67, 176, 197, 248] [14, 26, 51, 117, 144, 177, 209] [27, 52, 64, 69, 145, 178, 249] [16, 53, 66, 124, 179, 212, 248] [25, 45, 54, 118, 147, 180, 201] [27, 46, 55, 74, 146, 149, 181] [0, 47, 56, 120, 150, 182, 204] [48, 57, 76, 124, 126, 147, 183] [49, 58, 77, 90, 148, 152, 184] [50, 59, 75, 91, 153, 185, 206] [51, 60, 79, 92, 150, 186, 217] [20, 52, 61, 93, 151, 187, 250] [21, 53, 62, 95, 153, 188, 220] [16, 36, 63, 122, 155, 189, 210] [18, 37, 64, 123, 154, 157, 190] [19, 38, 65, 82, 158, 191, 213] [39, 66, 86, 121, 127, 155, 192] [40, 67, 84, 122, 156, 160, 193] [41, 68, 81, 85, 161, 194, 215] [28, 42, 69, 123, 158, 195, 225] [43, 70, 82, 87, 159, 196, 251] [30, 44, 71, 84, 161, 197, 228] [26, 41, 63, 72, 163, 198, 218] [43, 64, 73, 91, 162, 165, 199] [2, 65, 74, 124, 166, 200, 221] [44, 66, 75, 93, 128, 163, 201] [3, 67, 76, 94, 164, 168, 202] [4, 68, 77, 92, 169, 203, 223] [69, 78, 96, 106, 130, 166, 204] [36, 70, 79, 107, 167, 170, 205] [37, 71, 80, 109, 169, 206, 234] [30, 35, 54, 81, 172, 207, 226] [32, 55, 82, 125, 171, 174, 208] [33, 56, 83, 98, 175, 209, 229] [34, 57, 84, 102, 131, 172, 210] [35, 58, 85, 100, 173, 177, 211] [1, 59, 86, 101, 178, 212, 231] [45, 60, 87, 125, 133, 175, 213] [61, 88, 98, 103, 176, 179, 214] [47, 62, 89, 100, 178, 215, 240] [4, 61, 82, 90, 180, 183, 216] [8, 44, 83, 91, 184, 217, 235] [62, 84, 92, 107, 134, 181, 218] [9, 85, 93, 108, 182, 186, 219] [10, 86, 94, 106, 187, 220, 237] [11, 87, 95, 110, 136, 184, 221] [12, 54, 88, 96, 185, 188, 222] [55, 89, 97, 118, 139, 187, 223] [49, 53, 73, 98, 189, 192, 224] [6, 50, 74, 99, 193, 225, 241] [51, 75, 100, 114, 140, 190, 226] [52, 76, 101, 112, 191, 195, 227] [5, 77, 102, 113, 196, 228, 243] [53, 63, 78, 103, 142, 193, 229] [6, 79, 104, 115, 194, 197, 230] [65, 80, 105, 112, 145, 196, 231] [12, 80, 100, 106, 146, 199, 232] [13, 21, 101, 107, 200, 204, 233] [11, 22, 102, 108, 205, 234, 246] [23, 103, 109, 119, 148, 202, 235] [24, 72, 104, 110, 203, 206, 236] [26, 73, 105, 111, 151, 205, 237] [19, 69, 92, 112, 154, 208, 238] [17, 70, 93, 113, 209, 213, 239] [14, 18, 94, 114, 214, 240, 249] [71, 81, 95, 115, 156, 211, 241] [15, 96, 116, 121, 212, 215, 242] [17, 83, 97, 117, 159, 214, 243] [27, 39, 115, 118, 164, 219, 244] [0, 40, 116, 119, 220, 223, 245] [42, 90, 117, 120, 167, 222, 246] [1, 89, 109, 121, 173, 227, 247] [29, 34, 110, 122, 228, 231, 248] [31, 99, 111, 123, 176, 230, 249] [3, 60, 123, 124, 185, 236, 250] [7, 48, 120, 125, 194, 242, 251]

H_Z (126 checks, sparse supports)

[0, 9, 57, 126, 131, 182, 245] [1, 15, 66, 127, 128, 212, 247] [2, 21, 75, 128, 140, 153, 200] [3, 20, 23, 129, 141, 202, 250] [4, 24, 78, 130, 142, 203, 216] [5, 29, 84, 131, 134, 160, 228] [6, 28, 31, 132, 135, 225, 230] [7, 32, 87, 133, 136, 171, 251] [8, 37, 92, 134, 154, 169, 217] [9, 36, 39, 135, 155, 170, 219] [10, 40, 95, 129, 136, 156, 220] [2, 11, 37, 137, 157, 221, 234] [12, 38, 42, 138, 158, 222, 232] [13, 43, 97, 139, 159, 162, 233] [14, 46, 100, 140, 146, 177, 240] [15, 45, 48, 133, 141, 147, 242] [16, 49, 103, 142, 148, 179, 189] [5, 17, 46, 143, 149, 239, 243] [18, 47, 51, 144, 150, 190, 240] [19, 52, 105, 145, 151, 191, 238] [20, 55, 106, 130, 146, 171, 187] [21, 54, 57, 147, 172, 188, 233] [22, 58, 109, 135, 148, 173, 234] [8, 23, 55, 139, 149, 174, 235] [24, 56, 60, 137, 150, 175, 236] [25, 61, 111, 138, 151, 176, 180] [10, 26, 58, 152, 177, 198, 237] [27, 59, 62, 153, 178, 181, 244] [28, 64, 112, 145, 154, 162, 195] [29, 63, 66, 142, 155, 163, 248] [30, 67, 115, 156, 164, 197, 207] [14, 31, 64, 144, 157, 165, 249] [32, 65, 69, 145, 158, 166, 208] [33, 70, 117, 143, 159, 167, 209] [16, 34, 67, 160, 168, 210, 248] [35, 68, 71, 161, 169, 207, 211] [11, 36, 73, 136, 162, 189, 205] [37, 72, 75, 138, 163, 190, 206] [38, 76, 118, 139, 147, 164, 191] [20, 39, 73, 151, 165, 192, 244] [40, 74, 78, 149, 166, 193, 245] [41, 79, 120, 150, 167, 194, 198] [22, 42, 76, 126, 168, 195, 246] [43, 77, 80, 152, 169, 196, 199] [25, 44, 79, 170, 197, 201, 217] [17, 45, 82, 159, 171, 180, 213] [46, 81, 84, 156, 161, 172, 181] [47, 85, 121, 127, 173, 182, 215] [28, 48, 82, 158, 174, 183, 251] [49, 83, 87, 159, 175, 184, 224] [50, 88, 123, 157, 176, 185, 225] [30, 51, 85, 161, 177, 186, 226] [52, 86, 89, 127, 178, 187, 227] [33, 53, 88, 179, 188, 224, 229] [23, 54, 90, 148, 180, 207, 222] [0, 55, 92, 150, 181, 208, 223] [26, 56, 93, 151, 163, 182, 209] [36, 57, 90, 152, 167, 183, 210] [58, 91, 95, 153, 165, 184, 211] [59, 96, 124, 126, 166, 185, 212] [38, 60, 93, 128, 186, 213, 250] [61, 94, 97, 168, 187, 214, 216] [41, 62, 96, 130, 188, 215, 218] [31, 63, 98, 176, 189, 198, 229] [1, 64, 100, 173, 178, 190, 199] [34, 65, 101, 131, 191, 200, 231] [45, 66, 98, 175, 179, 192, 201] [67, 99, 103, 132, 176, 193, 202] [68, 104, 125, 133, 174, 194, 203] [47, 69, 101, 178, 195, 204, 238] [70, 102, 105, 131, 196, 205, 239] [50, 71, 104, 132, 197, 206, 241] [3, 39, 72, 127, 164, 198, 236] [2, 73, 106, 166, 199, 224, 237] [42, 74, 107, 167, 181, 200, 225] [3, 54, 75, 168, 185, 201, 226] [4, 76, 109, 169, 183, 202, 227] [44, 77, 110, 128, 184, 203, 228] [56, 78, 107, 134, 170, 204, 229] [79, 108, 111, 129, 186, 205, 230] [59, 80, 110, 136, 206, 231, 232] [6, 48, 81, 126, 194, 207, 241] [5, 82, 112, 191, 196, 208, 216] [51, 83, 113, 140, 209, 217, 243] [6, 63, 84, 193, 197, 210, 218] [7, 85, 115, 141, 194, 211, 219] [53, 86, 116, 142, 192, 212, 220] [65, 87, 113, 143, 196, 213, 221] [88, 114, 117, 140, 144, 214, 222] [68, 89, 116, 141, 215, 223, 247] [8, 11, 90, 132, 184, 216, 246] [12, 60, 91, 133, 185, 199, 217] [9, 72, 92, 186, 203, 218, 238] [10, 93, 118, 187, 201, 219, 239] [62, 94, 119, 134, 202, 220, 240] [12, 74, 95, 146, 188, 221, 241] [13, 96, 120, 135, 204, 222, 242] [77, 97, 119, 137, 148, 223, 243] [14, 17, 98, 129, 209, 214, 224] [18, 69, 99, 130, 154, 225, 249] [15, 81, 100, 211, 215, 226, 232] [16, 101, 121, 155, 212, 227, 233] [71, 102, 122, 156, 210, 228, 234] [18, 83, 103, 157, 214, 229, 235] [19, 104, 123, 154, 158, 230, 236] [86, 105, 122, 155, 160, 231, 237] [0, 21, 106, 143, 204, 220, 232] [22, 26, 107, 144, 205, 218, 233] [27, 80, 108, 145, 146, 219, 234] [24, 91, 109, 162, 206, 235, 247] [25, 110, 124, 147, 221, 236, 248] [27, 94, 111, 149, 164, 237, 249] [1, 29, 112, 137, 227, 231, 238] [30, 34, 113, 138, 172, 228, 239] [35, 89, 114, 139, 173, 226, 240] [32, 99, 115, 174, 230, 241, 244] [33, 116, 125, 171, 175, 242, 245] [35, 102, 117, 172, 177, 243, 246] [4, 40, 118, 160, 180, 223, 244] [41, 44, 119, 161, 163, 235, 245] [43, 108, 120, 165, 182, 246, 251] [7, 49, 121, 152, 192, 242, 247] [50, 53, 122, 153, 189, 193, 248] [52, 114, 123, 190, 195, 249, 250] [13, 61, 124, 179, 183, 200, 250] [19, 70, 125, 170, 208, 213, 251]

Code ID 252-30-10 · download JSON · raw on GitHub