[[234,82,3]] · QEC Challenge

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[[234,82,3]] d =

234

kd²/n

3.154

X/Z

0.0038

5.3852

layers

swaps

694

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)

[134, 142, 150]

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)

[17, 23, 29]

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)×234 (2,4)×936 (3,0)×117 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 234 (2,4): 936 (2,6): 1404 (3,0): 117 (3,4): 3627 (3,6): 11778 (3,8): 11232

trapping sets H_Z (1,4)×234 (2,4)×936 (3,0)×117 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 234 (2,4): 936 (2,6): 1404 (3,0): 117 (3,4): 3627 (3,6): 11778 (3,8): 11232

witness diameter X 2.2361 · 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

X checkZ checkqubit site (119)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 694 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 C3 x (C13 : C3) = SmallGroup(117, 3) (order 117), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 234; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 2, 5}, b = {0, 2, 9, 16} 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

[[234,82,3]] — two-block group-algebra code over C3 x (C13 : C3)

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| = 234

Parameters are also spelled out in provenance.construction of codes/234-82-3.json.

Parity checks

X-checks 117 (max weight 8) · Z-checks 117 (max weight 8)

H_X (117 checks, sparse supports)

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

H_Z (117 checks, sparse supports)

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

Code ID 234-82-3 · download JSON · raw on GitHub