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[[254,14,21]] d ≤
n
254
k
14
d
21
kd²/n
24.307
w
7
X/Z
1

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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)
[4, 12, 20, 27, 83, 99, 114, 122, 144, 145, 168, 169, 170, 189, 191, 192, 193, 210, 211, 213, 231]
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)
[20, 25, 29, 33, 53, 54, 75, 86, 107, 136, 140, 144, 147, 157, 180, 188, 192, 198, 226, 227, 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 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)×127 (2,4)×381 (3,3)×127 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 127 (1,4): 127 (2,4): 381 (2,5): 1524 (2,6): 762 (3,3): 127 (3,5): 1143 (3,6): 13462 (3,7): 19939 (3,8): 6858 (3,9): 2286 (3,10): 508
trapping sets H_Z (1,3)×127 (2,4)×381 (3,3)×127 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 127 (1,4): 127 (2,4): 381 (2,5): 1524 (2,6): 762 (3,3): 127 (3,5): 1143 (3,6): 13462 (3,7): 19939 (3,8): 6858 (3,9): 2286 (3,10): 508

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C127 = SmallGroup(127, 1) (order 127), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 254; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 46, 107}, b = {0, 8, 56} 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

[[254,14,21]] — two-block group-algebra code over C127

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| = 254

Parameters are also spelled out in provenance.construction of codes/254-14-21.json.

Parity checks

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