[[248,10,21]] · QEC Challenge

[[248,10,21]] d ≤

248

kd²/n

17.782

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 21, d_Z ≤ 21 · 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 21 · witness weight 21 (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, 21 qubits)

[7, 10, 11, 28, 47, 51, 72, 75, 91, 123, 145, 149, 164, 168, 178, 187, 206, 219, 225, 235, 241]

d_Z 21 · witness weight 21 (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, 21 qubits)

[11, 15, 27, 30, 36, 46, 62, 66, 68, 78, 96, 132, 154, 167, 177, 185, 190, 199, 207, 220, 228]

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)×124 (2,4)×372 (3,3)×124 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 124 (1,4): 124 (2,4): 372 (2,5): 1488 (2,6): 744 (3,3): 124 (3,5): 1364 (3,6): 13268 (3,7): 18724 (3,8): 6324 (3,9): 2232 (3,10): 496

trapping sets H_Z (1,3)×124 (2,4)×372 (3,3)×124 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 124 (1,4): 124 (2,4): 372 (2,5): 1488 (2,6): 744 (3,3): 124 (3,5): 1364 (3,6): 13268 (3,7): 18724 (3,8): 6324 (3,9): 2232 (3,10): 496

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C124 = SmallGroup(124, 2) (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, 17, 36, 68}, b = {0, 16, 38} 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

[[248,10,21]] — two-block group-algebra code over C124

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/248-10-21.json.

Parity checks

X-checks 124 (max weight 7) · Z-checks 124 (max weight 7)

H_X (124 checks, sparse supports)

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

H_Z (124 checks, sparse supports)

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

Code ID 248-10-21 · download JSON · raw on GitHub