[[120,44,3]] · QEC Challenge

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[[120,44,3]] d =

120

kd²/n

3.3

X/Z

0.0053

5.0

layers

swaps

262

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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)

[4, 11, 21]

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)

[104, 113, 118]

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)×120 (2,4)×480 (3,0)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 120 (2,4): 480 (2,6): 720 (3,0): 40 (3,4): 1800 (3,6): 6240 (3,8): 6000

trapping sets H_Z (1,4)×120 (2,4)×480 (3,0)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 120 (2,4): 480 (2,6): 720 (3,0): 40 (3,4): 1800 (3,6): 6240 (3,8): 6000

witness diameter X 1.0 · Z 1.0 (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 (61)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 262 nearest-neighbor SWAPs per round in total, at most 6 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 C15 : C4 = SmallGroup(60, 7) (order 60), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 120; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 3, 6}, b = {0, 3, 4, 11} 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

[[120,44,3]] — two-block group-algebra code over C15 : C4

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| = 120

Parameters are also spelled out in provenance.construction of codes/120-44-3.json.

Parity checks

X-checks 60 (max weight 8) · Z-checks 60 (max weight 8)

H_X (60 checks, sparse supports)

[0, 5, 10, 13, 60, 70, 95, 114] [0, 1, 6, 10, 61, 75, 101, 117] [1, 2, 6, 18, 62, 78, 104, 118] [0, 3, 13, 24, 60, 63, 95, 106] [4, 14, 21, 25, 60, 64, 70, 81] [2, 5, 13, 18, 65, 84, 109, 119] [0, 3, 6, 15, 61, 66, 101, 111] [4, 7, 16, 21, 61, 67, 75, 87] [2, 6, 8, 15, 62, 68, 104, 113] [7, 9, 16, 30, 62, 69, 78, 90] [3, 5, 10, 24, 63, 70, 106, 114] [4, 11, 25, 36, 60, 63, 64, 71] [12, 26, 33, 37, 64, 72, 81, 93] [2, 8, 13, 24, 65, 73, 109, 116] [9, 14, 25, 30, 65, 74, 84, 96] [1, 3, 10, 15, 66, 75, 111, 117] [4, 11, 16, 27, 61, 66, 67, 76] [12, 17, 28, 33, 67, 77, 87, 99] [1, 8, 15, 18, 68, 78, 113, 118] [9, 16, 19, 27, 62, 68, 69, 79] [17, 20, 28, 42, 69, 80, 90, 102] [11, 14, 21, 36, 63, 70, 71, 81] [12, 22, 37, 47, 64, 71, 72, 82] [23, 38, 45, 48, 72, 83, 93, 105] [5, 8, 18, 24, 73, 84, 116, 119] [9, 19, 25, 36, 65, 73, 74, 85] [20, 26, 37, 42, 74, 86, 96, 107] [7, 11, 21, 27, 66, 75, 76, 87] [12, 22, 28, 39, 67, 76, 77, 88] [23, 29, 40, 45, 77, 89, 99, 110] [7, 19, 27, 30, 68, 78, 79, 90] [20, 28, 31, 39, 69, 79, 80, 91] [29, 32, 40, 52, 80, 92, 102, 112] [22, 26, 33, 47, 71, 81, 82, 93] [23, 34, 48, 55, 72, 82, 83, 94] [35, 49, 54, 56, 83, 95, 105, 114] [14, 19, 30, 36, 73, 84, 85, 96] [20, 31, 37, 47, 74, 85, 86, 97] [32, 38, 48, 52, 86, 98, 107, 115] [17, 22, 33, 39, 76, 87, 88, 99] [23, 34, 40, 50, 77, 88, 89, 100] [35, 41, 51, 54, 89, 101, 110, 117] [17, 31, 39, 42, 79, 90, 91, 102] [32, 40, 43, 50, 80, 91, 92, 103] [41, 44, 51, 58, 92, 104, 112, 118] [34, 38, 45, 55, 82, 93, 94, 105] [35, 46, 56, 59, 83, 94, 95, 106] [26, 31, 42, 47, 85, 96, 97, 107] [32, 43, 48, 55, 86, 97, 98, 108] [44, 49, 56, 58, 98, 109, 115, 119] [29, 34, 45, 50, 88, 99, 100, 110] [35, 46, 51, 57, 89, 100, 101, 111] [29, 43, 50, 52, 91, 102, 103, 112] [44, 51, 53, 57, 92, 103, 104, 113] [46, 49, 54, 59, 94, 105, 106, 114] [38, 43, 52, 55, 97, 107, 108, 115] [44, 53, 56, 59, 98, 108, 109, 116] [41, 46, 54, 57, 100, 110, 111, 117] [41, 53, 57, 58, 103, 112, 113, 118] [49, 53, 58, 59, 108, 115, 116, 119]

H_Z (60 checks, sparse supports)

[0, 3, 4, 11, 60, 61, 63, 66] [1, 6, 7, 16, 61, 62, 75, 78] [2, 8, 9, 19, 62, 65, 68, 73] [3, 10, 11, 21, 63, 66, 70, 75] [4, 11, 12, 22, 64, 67, 71, 76] [5, 13, 14, 25, 60, 65, 70, 84] [6, 15, 16, 27, 61, 62, 66, 68] [7, 16, 17, 28, 67, 69, 87, 90] [8, 18, 19, 30, 68, 73, 78, 84] [9, 19, 20, 31, 69, 74, 79, 85] [0, 4, 10, 21, 60, 61, 70, 75] [11, 21, 22, 33, 71, 76, 81, 87] [12, 22, 23, 34, 72, 77, 82, 88] [13, 24, 25, 36, 60, 63, 65, 73] [14, 25, 26, 37, 64, 74, 81, 96] [1, 7, 15, 27, 66, 68, 75, 78] [16, 27, 28, 39, 67, 69, 76, 79] [17, 28, 29, 40, 77, 80, 99, 102] [2, 9, 18, 30, 62, 65, 78, 84] [19, 30, 31, 42, 79, 85, 90, 96] [20, 31, 32, 43, 80, 86, 91, 97] [4, 12, 21, 33, 64, 67, 81, 87] [22, 33, 34, 45, 82, 88, 93, 99] [23, 34, 35, 46, 83, 89, 94, 100] [5, 14, 24, 36, 63, 70, 73, 84] [25, 36, 37, 47, 64, 71, 74, 85] [26, 37, 38, 48, 72, 86, 93, 107] [7, 17, 27, 39, 76, 79, 87, 90] [28, 39, 40, 50, 77, 80, 88, 91] [29, 40, 41, 51, 89, 92, 110, 112] [9, 20, 30, 42, 69, 74, 90, 96] [31, 42, 43, 52, 91, 97, 102, 107] [32, 43, 44, 53, 92, 98, 103, 108] [12, 23, 33, 45, 72, 77, 93, 99] [34, 45, 46, 54, 94, 100, 105, 110] [0, 3, 35, 46, 95, 101, 106, 111] [14, 26, 36, 47, 71, 81, 85, 96] [37, 47, 48, 55, 72, 82, 86, 97] [38, 48, 49, 56, 83, 98, 105, 115] [17, 29, 39, 50, 88, 91, 99, 102] [40, 50, 51, 57, 89, 92, 100, 103] [1, 6, 41, 51, 101, 104, 117, 118] [20, 32, 42, 52, 80, 86, 102, 107] [43, 52, 53, 58, 103, 108, 112, 115] [2, 8, 44, 53, 104, 109, 113, 116] [23, 35, 45, 54, 83, 89, 105, 110] [3, 10, 46, 54, 106, 111, 114, 117] [26, 38, 47, 55, 82, 93, 97, 107] [48, 55, 56, 59, 83, 94, 98, 108] [5, 13, 49, 56, 95, 109, 114, 119] [29, 41, 50, 57, 100, 103, 110, 112] [6, 15, 51, 57, 101, 104, 111, 113] [32, 44, 52, 58, 92, 98, 112, 115] [8, 18, 53, 58, 113, 116, 118, 119] [0, 10, 35, 54, 95, 101, 114, 117] [38, 49, 55, 59, 94, 105, 108, 115] [13, 24, 56, 59, 95, 106, 109, 116] [1, 15, 41, 57, 111, 113, 117, 118] [2, 18, 44, 58, 104, 109, 118, 119] [5, 24, 49, 59, 106, 114, 116, 119]

Code ID 120-44-3 · download JSON · raw on GitHub