[[210,90,3]] · QEC Challenge

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[[210,90,3]] d =

210

kd²/n

3.857

X/Z

0.0033

5.831

layers

swaps

580

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 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)

[44, 102, 149]

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)

[42, 147, 184]

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 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)×105 (2,3)×210 (3,0)×105 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 105 (1,6): 105 (2,3): 210 (2,4): 315 (2,6): 105 (2,7): 1260 (2,8): 315 (2,10): 630 (3,0): 105 (3,3): 630 (3,4): 1260 (3,6): 4305 (3,7): 6405 (3,8): 3780 (3,9): 4515 (3,10): 5670 (3,11): 8190 (3,13): 630 (3,14): 4515

trapping sets H_Z (1,3)×105 (2,3)×210 (3,0)×105 (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 4.1231 · Z 3.6056 (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 (112)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 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 C5 x (C7 : C3) = SmallGroup(105, 1) (order 105), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 210; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 12, 23, 44, 81}, b = {0, 3, 44} 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 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

[[210,90,3]] — two-block group-algebra code over C5 x (C7 : C3)

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| = 210

Parameters are also spelled out in provenance.construction of codes/210-90-3.json.

Parity checks

X-checks 105 (max weight 9) · Z-checks 105 (max weight 9)

H_X (105 checks, sparse supports)

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

H_Z (105 checks, sparse supports)

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

Code ID 210-90-3 · download JSON · raw on GitHub