[[252,84,4]] · QEC Challenge

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[[252,84,4]] d =

252

kd²/n

5.333

X/Z

0.0052

5.6569

layers

swaps

865

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 4 · witness weight 4 (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, 4 qubits)

[4, 13, 184, 203]

d_Z 4 · witness weight 4 (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, 4 qubits)

[72, 91, 130, 139]

certificate exact, d = 4 · CryptoMiniSat 5.14.7 SAT

X: no logical < 4 exists; Z: no logical < 4 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 9 · H_Z 9

qubit degrees H_X 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)

trapping sets H_X (1,3)×126 (2,2)×126 (3,2)×378 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 126 (1,6): 126 (2,2): 126 (2,3): 126 (2,4): 252 (2,5): 630 (2,7): 630 (2,8): 504 (2,10): 378 (3,2): 378 (3,3): 1176 (3,4): 1134 (3,5): 1134 (3,6): 6342 (3,7): 3024 (3,8): 1512 (3,9): 9702 (3,10): 1890 (3,11): 1890 (3,12): 3780 (3,14): 1134

trapping sets H_Z (1,3)×126 (2,2)×126 (3,2)×378 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

witness diameter X 3.6056 · Z 4.4721 (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

X checkZ checkqubit site (140)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 865 nearest-neighbor SWAPs per round in total, at most 8 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 C126 = SmallGroup(126, 6) (order 126), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 252; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 4, 13, 39, 58, 86}, b = {0, 4, 19} as element indices. Check weight 9. 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 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

[[252,84,4]] — two-block group-algebra code over C126

Direction & hypothesis

Cell weight-9plus 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| = 252

Parameters are also spelled out in provenance.construction of codes/252-84-4.json.

Parity checks

X-checks 126 (max weight 9) · Z-checks 126 (max weight 9)

H_X (126 checks, sparse supports)

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

H_Z (126 checks, sparse supports)

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

Code ID 252-84-4 · download JSON · raw on GitHub