[[292,148,2]] · QEC Challenge

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[[292,148,2]] d =

292

148

kd²/n

2.027

X/Z

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (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, 2 qubits)

[224, 225]

d_Z 2 · witness weight 2 (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, 2 qubits)

[202, 203]

certificate exact, d = 2 · CryptoMiniSat 5.14.7 SAT

X: no logical < 2 exists; Z: no logical < 2 exists

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 4 · H_Z 4 (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)×292 (2,0)×146 (3,4)×12264 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 292 (2,0): 146 (2,4): 1752 (3,4): 12264 (3,8): 2336

trapping sets H_Z (1,4)×292 (2,0)×146 (3,4)×12264 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 292 (2,0): 146 (2,4): 1752 (3,4): 12264 (3,8): 2336

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C146 = SmallGroup(146, 2) (order 146), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 292; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 60, 61}, b = {0, 1, 48, 49} 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

[[292,148,2]] — two-block group-algebra code over C146

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| = 292

Parameters are also spelled out in provenance.construction of codes/292-148-2.json.

Parity checks

X-checks 146 (max weight 8) · Z-checks 146 (max weight 8)

H_X (146 checks, sparse supports)

[0, 1, 86, 87, 146, 147, 244, 245] [0, 1, 86, 87, 146, 147, 244, 245] [2, 3, 88, 89, 148, 149, 246, 247] [2, 3, 88, 89, 148, 149, 246, 247] [4, 5, 90, 91, 150, 151, 248, 249] [4, 5, 90, 91, 150, 151, 248, 249] [6, 7, 92, 93, 152, 153, 250, 251] [6, 7, 92, 93, 152, 153, 250, 251] [8, 9, 94, 95, 154, 155, 252, 253] [8, 9, 94, 95, 154, 155, 252, 253] [10, 11, 96, 97, 156, 157, 254, 255] [10, 11, 96, 97, 156, 157, 254, 255] [12, 13, 98, 99, 158, 159, 256, 257] [12, 13, 98, 99, 158, 159, 256, 257] [14, 15, 100, 101, 160, 161, 258, 259] [14, 15, 100, 101, 160, 161, 258, 259] [16, 17, 102, 103, 162, 163, 260, 261] [16, 17, 102, 103, 162, 163, 260, 261] [18, 19, 104, 105, 164, 165, 262, 263] [18, 19, 104, 105, 164, 165, 262, 263] [20, 21, 106, 107, 166, 167, 264, 265] [20, 21, 106, 107, 166, 167, 264, 265] [22, 23, 108, 109, 168, 169, 266, 267] [22, 23, 108, 109, 168, 169, 266, 267] [24, 25, 110, 111, 170, 171, 268, 269] [24, 25, 110, 111, 170, 171, 268, 269] [26, 27, 112, 113, 172, 173, 270, 271] [26, 27, 112, 113, 172, 173, 270, 271] [28, 29, 114, 115, 174, 175, 272, 273] [28, 29, 114, 115, 174, 175, 272, 273] [30, 31, 116, 117, 176, 177, 274, 275] [30, 31, 116, 117, 176, 177, 274, 275] [32, 33, 118, 119, 178, 179, 276, 277] [32, 33, 118, 119, 178, 179, 276, 277] [34, 35, 120, 121, 180, 181, 278, 279] [34, 35, 120, 121, 180, 181, 278, 279] [36, 37, 122, 123, 182, 183, 280, 281] [36, 37, 122, 123, 182, 183, 280, 281] [38, 39, 124, 125, 184, 185, 282, 283] [38, 39, 124, 125, 184, 185, 282, 283] [40, 41, 126, 127, 186, 187, 284, 285] [40, 41, 126, 127, 186, 187, 284, 285] [42, 43, 128, 129, 188, 189, 286, 287] [42, 43, 128, 129, 188, 189, 286, 287] [44, 45, 130, 131, 190, 191, 288, 289] [44, 45, 130, 131, 190, 191, 288, 289] [46, 47, 132, 133, 192, 193, 290, 291] [46, 47, 132, 133, 192, 193, 290, 291] [48, 49, 134, 135, 146, 147, 194, 195] [48, 49, 134, 135, 146, 147, 194, 195] [50, 51, 136, 137, 148, 149, 196, 197] [50, 51, 136, 137, 148, 149, 196, 197] [52, 53, 138, 139, 150, 151, 198, 199] [52, 53, 138, 139, 150, 151, 198, 199] [54, 55, 140, 141, 152, 153, 200, 201] [54, 55, 140, 141, 152, 153, 200, 201] [56, 57, 142, 143, 154, 155, 202, 203] [56, 57, 142, 143, 154, 155, 202, 203] [58, 59, 144, 145, 156, 157, 204, 205] [58, 59, 144, 145, 156, 157, 204, 205] [0, 1, 60, 61, 158, 159, 206, 207] [0, 1, 60, 61, 158, 159, 206, 207] [2, 3, 62, 63, 160, 161, 208, 209] [2, 3, 62, 63, 160, 161, 208, 209] [4, 5, 64, 65, 162, 163, 210, 211] [4, 5, 64, 65, 162, 163, 210, 211] [6, 7, 66, 67, 164, 165, 212, 213] [6, 7, 66, 67, 164, 165, 212, 213] [8, 9, 68, 69, 166, 167, 214, 215] [8, 9, 68, 69, 166, 167, 214, 215] [10, 11, 70, 71, 168, 169, 216, 217] [10, 11, 70, 71, 168, 169, 216, 217] [12, 13, 72, 73, 170, 171, 218, 219] [12, 13, 72, 73, 170, 171, 218, 219] [14, 15, 74, 75, 172, 173, 220, 221] [14, 15, 74, 75, 172, 173, 220, 221] [16, 17, 76, 77, 174, 175, 222, 223] [16, 17, 76, 77, 174, 175, 222, 223] [18, 19, 78, 79, 176, 177, 224, 225] [18, 19, 78, 79, 176, 177, 224, 225] [20, 21, 80, 81, 178, 179, 226, 227] [20, 21, 80, 81, 178, 179, 226, 227] [22, 23, 82, 83, 180, 181, 228, 229] [22, 23, 82, 83, 180, 181, 228, 229] [24, 25, 84, 85, 182, 183, 230, 231] [24, 25, 84, 85, 182, 183, 230, 231] [26, 27, 86, 87, 184, 185, 232, 233] [26, 27, 86, 87, 184, 185, 232, 233] [28, 29, 88, 89, 186, 187, 234, 235] [28, 29, 88, 89, 186, 187, 234, 235] [30, 31, 90, 91, 188, 189, 236, 237] [30, 31, 90, 91, 188, 189, 236, 237] [32, 33, 92, 93, 190, 191, 238, 239] [32, 33, 92, 93, 190, 191, 238, 239] [34, 35, 94, 95, 192, 193, 240, 241] [34, 35, 94, 95, 192, 193, 240, 241] [36, 37, 96, 97, 194, 195, 242, 243] [36, 37, 96, 97, 194, 195, 242, 243] [38, 39, 98, 99, 196, 197, 244, 245] [38, 39, 98, 99, 196, 197, 244, 245] [40, 41, 100, 101, 198, 199, 246, 247] [40, 41, 100, 101, 198, 199, 246, 247] [42, 43, 102, 103, 200, 201, 248, 249] [42, 43, 102, 103, 200, 201, 248, 249] [44, 45, 104, 105, 202, 203, 250, 251] [44, 45, 104, 105, 202, 203, 250, 251] [46, 47, 106, 107, 204, 205, 252, 253] [46, 47, 106, 107, 204, 205, 252, 253] [48, 49, 108, 109, 206, 207, 254, 255] [48, 49, 108, 109, 206, 207, 254, 255] [50, 51, 110, 111, 208, 209, 256, 257] [50, 51, 110, 111, 208, 209, 256, 257] [52, 53, 112, 113, 210, 211, 258, 259] [52, 53, 112, 113, 210, 211, 258, 259] [54, 55, 114, 115, 212, 213, 260, 261] [54, 55, 114, 115, 212, 213, 260, 261] [56, 57, 116, 117, 214, 215, 262, 263] [56, 57, 116, 117, 214, 215, 262, 263] [58, 59, 118, 119, 216, 217, 264, 265] [58, 59, 118, 119, 216, 217, 264, 265] [60, 61, 120, 121, 218, 219, 266, 267] [60, 61, 120, 121, 218, 219, 266, 267] [62, 63, 122, 123, 220, 221, 268, 269] [62, 63, 122, 123, 220, 221, 268, 269] [64, 65, 124, 125, 222, 223, 270, 271] [64, 65, 124, 125, 222, 223, 270, 271] [66, 67, 126, 127, 224, 225, 272, 273] [66, 67, 126, 127, 224, 225, 272, 273] [68, 69, 128, 129, 226, 227, 274, 275] [68, 69, 128, 129, 226, 227, 274, 275] [70, 71, 130, 131, 228, 229, 276, 277] [70, 71, 130, 131, 228, 229, 276, 277] [72, 73, 132, 133, 230, 231, 278, 279] [72, 73, 132, 133, 230, 231, 278, 279] [74, 75, 134, 135, 232, 233, 280, 281] [74, 75, 134, 135, 232, 233, 280, 281] [76, 77, 136, 137, 234, 235, 282, 283] [76, 77, 136, 137, 234, 235, 282, 283] [78, 79, 138, 139, 236, 237, 284, 285] [78, 79, 138, 139, 236, 237, 284, 285] [80, 81, 140, 141, 238, 239, 286, 287] [80, 81, 140, 141, 238, 239, 286, 287] [82, 83, 142, 143, 240, 241, 288, 289] [82, 83, 142, 143, 240, 241, 288, 289] [84, 85, 144, 145, 242, 243, 290, 291] [84, 85, 144, 145, 242, 243, 290, 291]

H_Z (146 checks, sparse supports)

[0, 1, 48, 49, 146, 147, 206, 207] [0, 1, 48, 49, 146, 147, 206, 207] [2, 3, 50, 51, 148, 149, 208, 209] [2, 3, 50, 51, 148, 149, 208, 209] [4, 5, 52, 53, 150, 151, 210, 211] [4, 5, 52, 53, 150, 151, 210, 211] [6, 7, 54, 55, 152, 153, 212, 213] [6, 7, 54, 55, 152, 153, 212, 213] [8, 9, 56, 57, 154, 155, 214, 215] [8, 9, 56, 57, 154, 155, 214, 215] [10, 11, 58, 59, 156, 157, 216, 217] [10, 11, 58, 59, 156, 157, 216, 217] [12, 13, 60, 61, 158, 159, 218, 219] [12, 13, 60, 61, 158, 159, 218, 219] [14, 15, 62, 63, 160, 161, 220, 221] [14, 15, 62, 63, 160, 161, 220, 221] [16, 17, 64, 65, 162, 163, 222, 223] [16, 17, 64, 65, 162, 163, 222, 223] [18, 19, 66, 67, 164, 165, 224, 225] [18, 19, 66, 67, 164, 165, 224, 225] [20, 21, 68, 69, 166, 167, 226, 227] [20, 21, 68, 69, 166, 167, 226, 227] [22, 23, 70, 71, 168, 169, 228, 229] [22, 23, 70, 71, 168, 169, 228, 229] [24, 25, 72, 73, 170, 171, 230, 231] [24, 25, 72, 73, 170, 171, 230, 231] [26, 27, 74, 75, 172, 173, 232, 233] [26, 27, 74, 75, 172, 173, 232, 233] [28, 29, 76, 77, 174, 175, 234, 235] [28, 29, 76, 77, 174, 175, 234, 235] [30, 31, 78, 79, 176, 177, 236, 237] [30, 31, 78, 79, 176, 177, 236, 237] [32, 33, 80, 81, 178, 179, 238, 239] [32, 33, 80, 81, 178, 179, 238, 239] [34, 35, 82, 83, 180, 181, 240, 241] [34, 35, 82, 83, 180, 181, 240, 241] [36, 37, 84, 85, 182, 183, 242, 243] [36, 37, 84, 85, 182, 183, 242, 243] [38, 39, 86, 87, 184, 185, 244, 245] [38, 39, 86, 87, 184, 185, 244, 245] [40, 41, 88, 89, 186, 187, 246, 247] [40, 41, 88, 89, 186, 187, 246, 247] [42, 43, 90, 91, 188, 189, 248, 249] [42, 43, 90, 91, 188, 189, 248, 249] [44, 45, 92, 93, 190, 191, 250, 251] [44, 45, 92, 93, 190, 191, 250, 251] [46, 47, 94, 95, 192, 193, 252, 253] [46, 47, 94, 95, 192, 193, 252, 253] [48, 49, 96, 97, 194, 195, 254, 255] [48, 49, 96, 97, 194, 195, 254, 255] [50, 51, 98, 99, 196, 197, 256, 257] [50, 51, 98, 99, 196, 197, 256, 257] [52, 53, 100, 101, 198, 199, 258, 259] [52, 53, 100, 101, 198, 199, 258, 259] [54, 55, 102, 103, 200, 201, 260, 261] [54, 55, 102, 103, 200, 201, 260, 261] [56, 57, 104, 105, 202, 203, 262, 263] [56, 57, 104, 105, 202, 203, 262, 263] [58, 59, 106, 107, 204, 205, 264, 265] [58, 59, 106, 107, 204, 205, 264, 265] [60, 61, 108, 109, 206, 207, 266, 267] [60, 61, 108, 109, 206, 207, 266, 267] [62, 63, 110, 111, 208, 209, 268, 269] [62, 63, 110, 111, 208, 209, 268, 269] [64, 65, 112, 113, 210, 211, 270, 271] [64, 65, 112, 113, 210, 211, 270, 271] [66, 67, 114, 115, 212, 213, 272, 273] [66, 67, 114, 115, 212, 213, 272, 273] [68, 69, 116, 117, 214, 215, 274, 275] [68, 69, 116, 117, 214, 215, 274, 275] [70, 71, 118, 119, 216, 217, 276, 277] [70, 71, 118, 119, 216, 217, 276, 277] [72, 73, 120, 121, 218, 219, 278, 279] [72, 73, 120, 121, 218, 219, 278, 279] [74, 75, 122, 123, 220, 221, 280, 281] [74, 75, 122, 123, 220, 221, 280, 281] [76, 77, 124, 125, 222, 223, 282, 283] [76, 77, 124, 125, 222, 223, 282, 283] [78, 79, 126, 127, 224, 225, 284, 285] [78, 79, 126, 127, 224, 225, 284, 285] [80, 81, 128, 129, 226, 227, 286, 287] [80, 81, 128, 129, 226, 227, 286, 287] [82, 83, 130, 131, 228, 229, 288, 289] [82, 83, 130, 131, 228, 229, 288, 289] [84, 85, 132, 133, 230, 231, 290, 291] [84, 85, 132, 133, 230, 231, 290, 291] [86, 87, 134, 135, 146, 147, 232, 233] [86, 87, 134, 135, 146, 147, 232, 233] [88, 89, 136, 137, 148, 149, 234, 235] [88, 89, 136, 137, 148, 149, 234, 235] [90, 91, 138, 139, 150, 151, 236, 237] [90, 91, 138, 139, 150, 151, 236, 237] [92, 93, 140, 141, 152, 153, 238, 239] [92, 93, 140, 141, 152, 153, 238, 239] [94, 95, 142, 143, 154, 155, 240, 241] [94, 95, 142, 143, 154, 155, 240, 241] [96, 97, 144, 145, 156, 157, 242, 243] [96, 97, 144, 145, 156, 157, 242, 243] [0, 1, 98, 99, 158, 159, 244, 245] [0, 1, 98, 99, 158, 159, 244, 245] [2, 3, 100, 101, 160, 161, 246, 247] [2, 3, 100, 101, 160, 161, 246, 247] [4, 5, 102, 103, 162, 163, 248, 249] [4, 5, 102, 103, 162, 163, 248, 249] [6, 7, 104, 105, 164, 165, 250, 251] [6, 7, 104, 105, 164, 165, 250, 251] [8, 9, 106, 107, 166, 167, 252, 253] [8, 9, 106, 107, 166, 167, 252, 253] [10, 11, 108, 109, 168, 169, 254, 255] [10, 11, 108, 109, 168, 169, 254, 255] [12, 13, 110, 111, 170, 171, 256, 257] [12, 13, 110, 111, 170, 171, 256, 257] [14, 15, 112, 113, 172, 173, 258, 259] [14, 15, 112, 113, 172, 173, 258, 259] [16, 17, 114, 115, 174, 175, 260, 261] [16, 17, 114, 115, 174, 175, 260, 261] [18, 19, 116, 117, 176, 177, 262, 263] [18, 19, 116, 117, 176, 177, 262, 263] [20, 21, 118, 119, 178, 179, 264, 265] [20, 21, 118, 119, 178, 179, 264, 265] [22, 23, 120, 121, 180, 181, 266, 267] [22, 23, 120, 121, 180, 181, 266, 267] [24, 25, 122, 123, 182, 183, 268, 269] [24, 25, 122, 123, 182, 183, 268, 269] [26, 27, 124, 125, 184, 185, 270, 271] [26, 27, 124, 125, 184, 185, 270, 271] [28, 29, 126, 127, 186, 187, 272, 273] [28, 29, 126, 127, 186, 187, 272, 273] [30, 31, 128, 129, 188, 189, 274, 275] [30, 31, 128, 129, 188, 189, 274, 275] [32, 33, 130, 131, 190, 191, 276, 277] [32, 33, 130, 131, 190, 191, 276, 277] [34, 35, 132, 133, 192, 193, 278, 279] [34, 35, 132, 133, 192, 193, 278, 279] [36, 37, 134, 135, 194, 195, 280, 281] [36, 37, 134, 135, 194, 195, 280, 281] [38, 39, 136, 137, 196, 197, 282, 283] [38, 39, 136, 137, 196, 197, 282, 283] [40, 41, 138, 139, 198, 199, 284, 285] [40, 41, 138, 139, 198, 199, 284, 285] [42, 43, 140, 141, 200, 201, 286, 287] [42, 43, 140, 141, 200, 201, 286, 287] [44, 45, 142, 143, 202, 203, 288, 289] [44, 45, 142, 143, 202, 203, 288, 289] [46, 47, 144, 145, 204, 205, 290, 291] [46, 47, 144, 145, 204, 205, 290, 291]

Code ID 292-148-2 · download JSON · raw on GitHub