[[248,24,18]] · QEC Challenge

[[248,24,18]] d ≤

248

kd²/n

31.355

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)

[13, 15, 32, 42, 52, 78, 85, 121, 132, 133, 134, 145, 147, 158, 164, 201, 220, 227]

d_Z 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)

[18, 35, 65, 85, 87, 98, 110, 111, 116, 120, 153, 164, 167, 203, 210, 217, 228, 236]

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)×1860 (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): 1860 (3,8): 67332 (3,10): 6944

trapping sets H_Z (1,4)×248 (2,6)×3472 (3,6)×1860 (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): 1860 (3,8): 67332 (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, 60, 98}, b = {0, 9, 18, 90} 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

[[248,24,18]] — 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/248-24-18.json.

Parity checks

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

H_X (124 checks, sparse supports)

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

H_Z (124 checks, sparse supports)

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

Code ID 248-24-18 · download JSON · raw on GitHub