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[[276,96,3]] d =
n
276
k
96
d
3
kd²/n
3.13
w
8
X/Z
1
g
0.0023
r
6.0828
layers
2
swaps
1170

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (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, 3 qubits)
[162, 169, 176]
d_Z 3 · witness weight 3 (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, 3 qubits)
[90, 97, 104]
certificate exact, d = 3 · CryptoMiniSat 5.14.7 SAT
X: no logical < 3 exists; Z: no logical < 3 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)×276 (2,4)×1104 (3,0)×92 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 276 (2,4): 1104 (2,6): 1656 (3,0): 92 (3,4): 4140 (3,6): 14352 (3,8): 13800
trapping sets H_Z (1,4)×276 (2,4)×1104 (3,0)×92 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 276 (2,4): 1104 (2,6): 1656 (3,0): 92 (3,4): 4140 (3,6): 14352 (3,8): 13800
witness diameter X 2.0 · Z 1.4142 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @MathysRennela · Simulated annealing over integer-grid site assignments with the repo's own research/local2d/fold_layout.search_layout, single layer at radius 4 then two layers at 7, plus a divisor scan of the two-block index convention Z_l x Z_m which gives the tighter embedding here; radius and per-site occupancy re-measured with the verifier's own rule. · 2026-10-13
r = 6.083
X checkZ checkqubit site (142)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 1170 nearest-neighbor SWAPs per round in total, at most 9 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C23 x S3 = SmallGroup(138, 1) (order 138), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 276; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 48, 55}, b = {0, 3, 63, 70} 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 2bga-nonabelian (a tag, not a ranking)
locality 2D-local bilayer (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

[[276,96,3]] — two-block group-algebra code over C23 x S3

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| = 276

Parameters are also spelled out in provenance.construction of codes/276-96-3.json.

Parity checks

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