[[204,72,3]] · QEC Challenge

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[[204,72,3]] d =

204

kd²/n

3.176

X/Z

0.0047

5.099

layers

swaps

580

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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)

[180, 187, 194]

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)

[106, 112, 119]

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)×204 (2,4)×816 (3,0)×68 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 204 (2,4): 816 (2,6): 1224 (3,0): 68 (3,4): 3060 (3,6): 10608 (3,8): 10200

trapping sets H_Z (1,4)×204 (2,4)×816 (3,0)×68 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 204 (2,4): 816 (2,6): 1224 (3,0): 68 (3,4): 3060 (3,6): 10608 (3,8): 10200

witness diameter X 2.2361 · Z 2.2361 (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 (110)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 580 nearest-neighbor SWAPs per round in total, at most 7 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 C17 x S3 = SmallGroup(102, 1) (order 102), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 204; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 21, 28}, b = {0, 3, 18, 25} 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

[[204,72,3]] — two-block group-algebra code over C17 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| = 204

Parameters are also spelled out in provenance.construction of codes/204-72-3.json.

Parity checks

X-checks 102 (max weight 8) · Z-checks 102 (max weight 8)

H_X (102 checks, sparse supports)

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

H_Z (102 checks, sparse supports)

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

Code ID 204-72-3 · download JSON · raw on GitHub