[[252,26,18]] · QEC Challenge

[[252,26,18]] d ≤

252

kd²/n

33.429

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 8, w_Z = 8 (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)

[0, 5, 29, 32, 45, 77, 82, 113, 143, 150, 163, 175, 184, 186, 223, 228, 242, 243]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 5000 trials · survived 5000 trials · 2026-10-12

witness operator (support, 18 qubits)

[13, 28, 32, 64, 73, 74, 87, 94, 114, 119, 145, 173, 176, 179, 188, 221, 225, 243]

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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×252 (2,6)×3528 (3,6)×1764 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 252 (2,6): 3528 (3,6): 1764 (3,8): 68796 (3,10): 7056

trapping sets H_Z (1,4)×252 (2,6)×3528 (3,6)×1764 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 252 (2,6): 3528 (3,6): 1764 (3,8): 68796 (3,10): 7056

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C7 x D18 = SmallGroup(126, 3) (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, 48, 102}, b = {0, 8, 75, 76} as element indices. Check weight 8. 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. The Z witness is the database's own published logical operator rather than one this session's search produced; that side's witness_provenance records the difference. 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

[[252,26,18]] — two-block group-algebra code over C7 x D18

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 5,000 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/252-26-18.json.

Parity checks

X-checks 126 (max weight 8) · Z-checks 126 (max weight 8)

H_X (126 checks, sparse supports)

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

H_Z (126 checks, sparse supports)

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

Code ID 252-26-18 · download JSON · raw on GitHub