[[252,20,16]] · QEC Challenge

[[252,20,16]] d ≤

252

kd²/n

20.317

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (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, 16 qubits)

[44, 69, 94, 111, 113, 119, 126, 129, 135, 137, 204, 220, 221, 222, 234, 237]

d_Z 16 · witness weight 16 (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, 16 qubits)

[7, 16, 48, 49, 66, 69, 114, 116, 122, 125, 142, 144, 153, 195, 237, 247]

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 (3,5)×1764 (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 (3,5): 1764 (3,6): 13398 (3,7): 19026 (3,8): 6678 (3,9): 2268 (3,10): 504

trapping sets H_Z (1,3)×126 (2,4)×378 (3,5)×1764 (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 (3,5): 1764 (3,6): 13398 (3,7): 19026 (3,8): 6678 (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, 3, 10, 79}, b = {0, 5, 29} 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

[[252,20,16]] — 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/252-20-16.json.

Parity checks

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

H_X (126 checks, sparse supports)

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

H_Z (126 checks, sparse supports)

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

Code ID 252-20-16 · download JSON · raw on GitHub