[[300,20,17]] · QEC Challenge

[[300,20,17]] d ≤

300

kd²/n

19.267

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (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, 17 qubits)

[10, 21, 41, 44, 51, 77, 157, 165, 200, 203, 209, 227, 232, 234, 237, 255, 295]

d_Z 17 · witness weight 17 (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, 17 qubits)

[4, 11, 21, 23, 24, 38, 46, 48, 67, 96, 99, 112, 115, 118, 120, 142, 143]

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)×150 (2,4)×450 (3,5)×1800 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 150 (1,4): 150 (2,4): 450 (2,5): 1800 (2,6): 900 (3,5): 1800 (3,6): 15900 (3,7): 23550 (3,8): 8100 (3,9): 2700 (3,10): 600

trapping sets H_Z (1,3)×150 (2,4)×450 (3,5)×1800 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 150 (1,4): 150 (2,4): 450 (2,5): 1800 (2,6): 900 (3,5): 1800 (3,6): 15900 (3,7): 23550 (3,8): 8100 (3,9): 2700 (3,10): 600

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C5 x D30 = SmallGroup(150, 11) (order 150), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 300; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 21, 27, 43}, b = {0, 5, 54} 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 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

[[300,20,17]] — two-block group-algebra code over C5 x D30

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| = 300

Parameters are also spelled out in provenance.construction of codes/300-20-17.json.

Parity checks

X-checks 150 (max weight 7) · Z-checks 150 (max weight 7)

H_X (150 checks, sparse supports)

[0, 64, 73, 93, 150, 200, 228] [1, 36, 51, 79, 151, 188, 215] [2, 88, 97, 114, 151, 152, 251] [3, 20, 28, 116, 153, 202, 250] [4, 91, 99, 117, 154, 201, 288] [5, 57, 75, 102, 150, 155, 239] [6, 59, 77, 104, 156, 190, 238] [7, 60, 78, 105, 157, 189, 282] [8, 11, 28, 118, 155, 158, 168] [9, 38, 46, 131, 159, 167, 226] [10, 112, 120, 132, 160, 211, 225] [11, 39, 48, 133, 161, 225, 253] [12, 40, 49, 134, 162, 179, 299] [11, 13, 20, 115, 163, 273, 290] [6, 14, 19, 81, 152, 162, 164] [15, 83, 100, 122, 161, 165, 213] [16, 84, 101, 123, 166, 199, 212] [17, 19, 29, 124, 167, 212, 241] [18, 86, 103, 125, 168, 170, 298] [19, 87, 104, 126, 169, 264, 284] [20, 23, 46, 133, 164, 170, 183] [0, 21, 48, 69, 157, 171, 182] [22, 26, 49, 135, 156, 172, 235] [23, 62, 71, 142, 156, 173, 271] [24, 63, 72, 143, 174, 200, 261] [23, 25, 38, 130, 175, 187, 296] [26, 65, 74, 144, 176, 227, 296] [0, 27, 39, 66, 177, 276, 295] [26, 28, 40, 133, 178, 252, 275] [15, 29, 34, 104, 158, 174, 179] [17, 30, 36, 106, 154, 173, 180] [18, 31, 37, 107, 153, 181, 222] [32, 34, 50, 136, 153, 182, 262] [33, 109, 121, 137, 183, 188, 249] [34, 110, 122, 138, 178, 184, 293] [35, 37, 52, 139, 185, 214, 293] [36, 111, 124, 140, 186, 267, 292] [6, 29, 37, 125, 187, 240, 266] [38, 41, 69, 142, 179, 188, 204] [2, 39, 71, 93, 166, 189, 203] [40, 44, 72, 144, 165, 190, 258] [3, 41, 73, 95, 165, 185, 191] [4, 42, 74, 96, 151, 192, 279] [0, 41, 43, 47, 193, 208, 237] [44, 89, 98, 148, 194, 237, 250] [2, 45, 62, 90, 195, 236, 288] [44, 46, 63, 142, 169, 196, 287] [3, 47, 64, 92, 197, 287, 297] [4, 48, 65, 93, 198, 255, 286] [47, 49, 66, 144, 199, 226, 254] [30, 50, 55, 122, 170, 192, 200] [32, 51, 57, 124, 160, 191, 201] [33, 52, 58, 125, 159, 202, 246] [1, 53, 55, 59, 159, 176, 203] [35, 54, 60, 127, 150, 204, 270] [36, 55, 61, 128, 196, 205, 224] [56, 58, 76, 145, 206, 224, 238] [57, 129, 136, 146, 207, 223, 282] [15, 50, 58, 137, 163, 208, 281] [59, 61, 79, 147, 209, 281, 294] [17, 51, 60, 139, 210, 243, 280] [18, 52, 61, 140, 211, 213, 242] [8, 62, 95, 114, 181, 212, 227] [63, 67, 96, 148, 180, 213, 276] [9, 64, 97, 116, 180, 206, 214] [10, 65, 98, 117, 155, 215, 291] [2, 64, 66, 70, 216, 232, 260] [12, 67, 99, 119, 167, 217, 260] [3, 8, 13, 68, 211, 218, 259] [4, 67, 69, 73, 184, 210, 219] [9, 70, 88, 113, 210, 220, 299] [10, 71, 89, 114, 209, 221, 273] [70, 72, 90, 148, 157, 222, 272] [12, 73, 91, 116, 223, 272, 289] [13, 74, 92, 117, 224, 229, 271] [53, 75, 81, 136, 172, 214, 225] [54, 76, 82, 137, 171, 226, 267] [5, 77, 79, 83, 171, 194, 227] [56, 78, 84, 139, 152, 228, 285] [57, 79, 85, 140, 219, 229, 248] [7, 80, 82, 86, 161, 230, 248] [59, 81, 87, 141, 199, 231, 247] [1, 30, 60, 82, 175, 198, 232] [83, 85, 102, 149, 198, 233, 298] [32, 75, 84, 145, 197, 234, 264] [33, 76, 85, 146, 154, 235, 263] [1, 77, 86, 105, 236, 263, 283] [35, 78, 87, 147, 216, 237, 262] [21, 88, 118, 131, 201, 230, 238] [22, 89, 119, 132, 164, 239, 297] [8, 88, 90, 94, 240, 255, 278] [24, 91, 120, 134, 182, 241, 278] [9, 20, 25, 92, 235, 242, 277] [10, 91, 93, 97, 205, 234, 243] [11, 21, 27, 94, 234, 244, 261] [12, 22, 28, 95, 187, 233, 245] [13, 94, 96, 99, 166, 186, 246] [24, 97, 112, 131, 186, 247, 295] [25, 98, 113, 132, 185, 248, 252] [27, 99, 115, 134, 249, 251, 274] [14, 100, 102, 106, 189, 217, 250] [80, 101, 107, 145, 158, 251, 294] [81, 102, 108, 146, 243, 252, 269] [16, 103, 105, 109, 173, 253, 269] [83, 104, 110, 147, 222, 254, 268] [5, 51, 84, 105, 193, 221, 255] [19, 106, 108, 111, 221, 249, 256] [6, 53, 86, 107, 178, 220, 257] [7, 54, 87, 108, 160, 177, 258] [5, 100, 109, 123, 177, 259, 292] [56, 101, 110, 149, 176, 240, 260] [7, 103, 111, 126, 239, 261, 265] [42, 112, 135, 143, 203, 262, 290] [21, 38, 43, 113, 258, 263, 289] [22, 112, 114, 118, 229, 257, 264] [23, 39, 45, 115, 257, 265, 279] [24, 40, 46, 116, 208, 256, 266] [25, 115, 117, 120, 181, 207, 267] [26, 42, 48, 118, 207, 236, 268] [27, 43, 49, 119, 169, 206, 269] [45, 120, 130, 143, 168, 270, 286] [31, 121, 123, 127, 191, 271, 284] [106, 122, 128, 149, 246, 272, 283] [14, 75, 107, 123, 216, 245, 273] [34, 124, 126, 129, 245, 270, 274] [15, 77, 109, 125, 196, 244, 275] [16, 78, 110, 126, 172, 195, 276] [14, 17, 37, 127, 195, 223, 277] [18, 80, 111, 128, 163, 194, 278] [16, 121, 129, 138, 162, 279, 280] [41, 62, 68, 130, 275, 280, 291] [42, 63, 69, 131, 232, 274, 281] [43, 130, 132, 135, 202, 231, 282] [44, 65, 71, 133, 231, 259, 283] [45, 66, 72, 134, 184, 230, 284] [47, 68, 74, 135, 183, 209, 285] [55, 136, 138, 141, 266, 285, 286] [30, 100, 127, 137, 219, 265, 287] [31, 101, 128, 138, 190, 218, 288] [29, 32, 58, 139, 218, 247, 289] [33, 103, 129, 140, 175, 217, 290] [31, 35, 61, 141, 174, 197, 291] [67, 89, 95, 142, 254, 277, 292] [68, 90, 96, 143, 205, 253, 293] [70, 92, 98, 144, 204, 233, 294] [50, 53, 82, 145, 242, 268, 295] [54, 121, 141, 146, 193, 241, 296] [52, 56, 85, 147, 192, 220, 297] [94, 113, 119, 148, 228, 256, 298] [76, 80, 108, 149, 215, 244, 299]

H_Z (150 checks, sparse supports)

[0, 5, 54, 150, 171, 177, 193] [1, 2, 42, 151, 203, 232, 236] [2, 14, 78, 152, 189, 195, 216] [3, 31, 32, 153, 191, 197, 218] [4, 30, 85, 154, 192, 198, 219] [5, 8, 65, 155, 227, 255, 259] [6, 22, 23, 156, 164, 187, 257] [7, 21, 72, 157, 230, 258, 261] [8, 29, 101, 158, 212, 218, 240] [9, 52, 53, 159, 214, 220, 242] [10, 51, 108, 160, 215, 221, 243] [11, 15, 80, 158, 161, 163, 244] [12, 14, 129, 162, 217, 223, 245] [13, 58, 128, 163, 218, 224, 246] [14, 20, 89, 164, 250, 273, 277] [15, 40, 41, 165, 179, 208, 275] [16, 39, 96, 166, 253, 276, 279] [9, 17, 67, 167, 180, 210, 277] [8, 18, 120, 168, 181, 211, 278] [19, 46, 119, 164, 167, 169, 256] [18, 20, 50, 153, 163, 170, 242] [21, 76, 77, 171, 238, 244, 263] [22, 75, 126, 172, 239, 245, 264] [23, 30, 103, 170, 173, 175, 265] [24, 29, 141, 174, 241, 247, 266] [25, 82, 140, 175, 242, 248, 267] [26, 53, 110, 172, 176, 178, 268] [27, 108, 109, 177, 244, 249, 269] [28, 34, 107, 153, 158, 178, 245] [12, 29, 38, 167, 179, 187, 289] [30, 63, 64, 180, 200, 232, 287] [31, 62, 117, 181, 271, 288, 291] [21, 32, 91, 182, 201, 234, 289] [20, 33, 135, 183, 202, 235, 290] [34, 69, 134, 179, 182, 184, 274] [35, 41, 98, 185, 204, 237, 291] [36, 96, 97, 151, 180, 186, 205] [25, 37, 95, 181, 185, 187, 277] [1, 33, 38, 159, 175, 188, 263] [7, 39, 100, 161, 177, 189, 265] [6, 40, 138, 162, 178, 190, 266] [41, 51, 121, 188, 191, 193, 280] [42, 50, 147, 192, 262, 268, 281] [43, 105, 146, 193, 263, 269, 282] [44, 77, 128, 190, 194, 196, 283] [45, 126, 127, 195, 265, 270, 284] [46, 55, 125, 159, 170, 196, 266] [47, 84, 141, 193, 197, 199, 285] [48, 82, 83, 161, 171, 198, 268] [16, 49, 81, 162, 172, 199, 269] [0, 24, 50, 182, 200, 208, 295] [4, 51, 88, 151, 201, 210, 255] [3, 52, 132, 185, 202, 211, 297] [39, 53, 112, 203, 225, 257, 295] [38, 54, 144, 204, 226, 258, 296] [55, 93, 143, 200, 203, 205, 286] [56, 64, 119, 206, 228, 260, 297] [57, 117, 118, 155, 201, 207, 229] [43, 58, 116, 202, 206, 208, 289] [59, 71, 135, 156, 203, 209, 231] [60, 69, 70, 157, 204, 210, 232] [10, 61, 68, 205, 209, 211, 291] [16, 17, 62, 173, 195, 212, 280] [15, 61, 63, 174, 196, 213, 281] [35, 64, 75, 150, 197, 214, 216] [1, 65, 149, 176, 198, 215, 283] [66, 87, 123, 177, 199, 216, 284] [67, 100, 140, 213, 217, 219, 292] [68, 138, 139, 218, 280, 285, 293] [69, 79, 137, 171, 188, 219, 281] [70, 107, 147, 216, 220, 222, 294] [71, 105, 106, 173, 189, 221, 283] [31, 72, 104, 174, 190, 222, 284] [57, 73, 127, 150, 191, 219, 223] [55, 56, 74, 176, 192, 224, 285] [10, 11, 75, 155, 225, 234, 273] [9, 49, 76, 206, 226, 235, 299] [26, 62, 77, 156, 227, 236, 275] [0, 78, 148, 157, 228, 237, 276] [74, 79, 114, 151, 209, 227, 229] [80, 88, 134, 230, 251, 278, 299] [81, 132, 133, 164, 225, 231, 252] [66, 82, 131, 226, 230, 232, 295] [83, 95, 144, 165, 227, 233, 254] [84, 93, 94, 166, 228, 234, 255] [22, 85, 92, 229, 233, 235, 297] [45, 86, 118, 168, 230, 236, 257] [43, 44, 87, 169, 231, 237, 258] [6, 56, 88, 152, 220, 238, 240] [5, 89, 111, 194, 221, 239, 292] [37, 90, 110, 195, 222, 240, 293] [17, 91, 146, 154, 223, 241, 243] [61, 92, 145, 197, 224, 242, 294] [60, 93, 102, 150, 189, 198, 243] [94, 125, 149, 240, 244, 246, 298] [95, 123, 124, 191, 212, 245, 292] [52, 96, 122, 192, 213, 246, 293] [81, 97, 139, 152, 214, 243, 247] [79, 80, 98, 194, 215, 248, 294] [33, 99, 106, 154, 217, 246, 249] [3, 44, 100, 165, 250, 259, 287] [2, 99, 101, 166, 251, 260, 288] [28, 98, 102, 155, 233, 250, 252] [11, 103, 143, 168, 253, 261, 290] [49, 104, 142, 156, 169, 179, 254] [48, 90, 105, 157, 236, 253, 255] [106, 116, 148, 180, 250, 256, 272] [107, 114, 115, 181, 251, 257, 273] [40, 108, 113, 252, 256, 258, 299] [68, 109, 133, 183, 253, 259, 275] [66, 67, 110, 184, 254, 260, 276] [24, 94, 111, 186, 256, 261, 278] [32, 87, 112, 160, 247, 262, 264] [85, 86, 113, 220, 248, 263, 298] [19, 84, 114, 152, 212, 221, 264] [111, 115, 137, 163, 249, 265, 267] [37, 116, 136, 153, 214, 223, 266] [36, 76, 117, 154, 215, 224, 267] [104, 118, 145, 158, 238, 264, 268] [102, 103, 119, 217, 239, 269, 298] [54, 120, 124, 160, 241, 267, 270] [23, 74, 121, 183, 271, 279, 296] [72, 73, 122, 165, 184, 200, 272] [13, 71, 123, 166, 259, 271, 273] [99, 124, 131, 167, 186, 201, 274] [28, 125, 130, 168, 187, 202, 275] [27, 63, 126, 169, 261, 274, 276] [92, 127, 142, 204, 271, 277, 287] [90, 91, 128, 205, 272, 278, 288] [42, 115, 129, 207, 274, 279, 290] [60, 129, 130, 175, 270, 280, 282] [58, 59, 131, 159, 238, 247, 281] [7, 57, 132, 160, 239, 248, 282] [86, 122, 133, 161, 170, 178, 283] [19, 121, 134, 162, 241, 249, 284] [78, 135, 136, 172, 262, 282, 285] [48, 120, 136, 182, 207, 225, 286] [46, 47, 137, 183, 208, 226, 287] [4, 45, 138, 184, 279, 286, 288] [73, 113, 139, 185, 210, 228, 289] [13, 112, 140, 186, 211, 229, 290] [65, 130, 141, 231, 286, 291, 296] [36, 109, 142, 173, 188, 196, 292] [34, 35, 143, 174, 262, 270, 293] [59, 101, 144, 176, 190, 199, 294] [27, 97, 145, 206, 234, 251, 295] [25, 26, 146, 207, 235, 252, 296] [47, 89, 147, 209, 237, 254, 297] [18, 83, 148, 194, 213, 222, 298] [12, 70, 149, 233, 260, 272, 299]

Code ID 300-20-17 · download JSON · raw on GitHub