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[[112,6,8]] d =
n
112
k
6
d
8
kd²/n
3.429
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 8 · witness weight 8 (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, 8 qubits)
[10, 13, 22, 27, 59, 65, 74, 88]
d_Z 8 · witness weight 8 (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, 8 qubits)
[14, 31, 36, 39, 66, 69, 78, 83]
certificate exact, d = 8 · CryptoMiniSat 5.14 SAT
X: no logical < 8 exists; Z: no logical < 8 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×56 (2,2)×56 (3,2)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 56 (1,3): 56 (2,2): 56 (2,3): 336 (2,4): 168 (3,2): 56 (3,3): 1064 (3,4): 1848 (3,5): 672 (3,6): 336 (3,7): 56
trapping sets H_Z (1,2)×56 (2,2)×56 (3,2)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 56 (1,3): 56 (2,2): 56 (2,3): 336 (2,4): 168 (3,2): 56 (3,3): 1064 (3,4): 1848 (3,5): 672 (3,6): 336 (3,7): 56

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C56 = SmallGroup(56, 2) (order 56), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 112; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 25}, b = {0, 1} as element indices. Check weight 5. 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 unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[112,6,8]] — two-block group-algebra code over C56

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| = 112

Parameters are also spelled out in provenance.construction of codes/112-6-8.json.

Parity checks

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