[[248,16,22]] · QEC Challenge

[[248,16,22]] d ≤

248

kd²/n

31.226

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 22, d_Z ≤ 22 · 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 22 · witness weight 22 (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, 22 qubits)

[19, 22, 25, 28, 33, 35, 36, 38, 117, 120, 133, 135, 136, 138, 193, 196, 219, 222, 227, 230, 233, 236]

d_Z 22 · witness weight 22 (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, 22 qubits)

[43, 48, 83, 88, 109, 110, 111, 116, 117, 118, 129, 130, 153, 154, 203, 208, 211, 216, 223, 228, 233, 234]

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)×248 (2,6)×3472 (3,6)×1984 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 248 (2,6): 3472 (3,6): 1984 (3,8): 66960 (3,10): 6944

trapping sets H_Z (1,4)×248 (2,6)×3472 (3,6)×1984 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 248 (2,6): 3472 (3,6): 1984 (3,8): 66960 (3,10): 6944

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C62 x C2 = SmallGroup(124, 4) (order 124), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 248; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 5, 40, 102}, b = {0, 16, 42, 85} 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. 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

[[248,16,22]] — two-block group-algebra code over C62 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| = 248

Parameters are also spelled out in provenance.construction of codes/248-16-22.json.

Parity checks

X-checks 124 (max weight 8) · Z-checks 124 (max weight 8)

H_X (124 checks, sparse supports)

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

H_Z (124 checks, sparse supports)

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

Code ID 248-16-22 · download JSON · raw on GitHub