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[[110,28,10]] d =
n
110
k
28
d
10
kd²/n
25.4545
w
10
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[14, 32, 41, 48, 58, 63, 72, 74, 76, 93]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[9, 42, 44, 47, 48, 51, 79, 82, 94, 96]
certificate exact, d = 10 · tropidist b03ec2d4077f (arXiv:2611.08247) exhaustive search · level solver: a solver verdict, no artifact a third party can check without re-running the encoding
X: no logical < 10 exists; Z: no logical < 10 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 10 · H_Z 10
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×110 (2,6)×1980 (3,6)×3520 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 110 (2,6): 1980 (3,6): 3520 (3,8): 42900 (3,10): 5280
trapping sets H_Z (1,4)×110 (2,6)×1980 (3,6)×3520 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 110 (2,6): 1980 (3,6): 3520 (3,8): 42900 (3,10): 5280

Circuit tier

syndrome-extraction memory circuits committed under circuits/110-28-10/ · canonical noise recipe, 10 rounds, stim 1.16.0
d_circ ≤ 10 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 10 · fault-set witness of 10 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 10)
[101085, 101086, 101101, 101104, 101114, 101117, 101121, 101127, 101165, 101167]
d_circ^Z 10 · fault-set witness of 10 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 10)
[101077, 101078, 101085, 101086, 101096, 101098, 101103, 101108, 101113, 101115]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Symplectically halved pair-partition code (Lee et al. 2026) converted to CSS via LC (arXiv:2610.28737).
model Beam 1.0 (claimed, not verified)
date 2026-10-22
family pair-partition CPM (a tag, not a ranking)
locality unrestricted (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

[[110,28,10]] — halved pair-partition code converted to CSS

Direction & hypothesis

Aim at the unrestricted / weight-9plus cell with a high-rate stabilizer code. The paper (arXiv:2610.28737) proves that among the 191 symplectically halved pair-partition entries of Lee et al. (arXiv:2609.30069) only one is LC-equivalent to CSS: the [[110,28,10]] plain-fold instance. Converting it to CSS should place a (110,28,10) code on the CSS board, where no CSS entry dominates it on (n,k,d,w).

What was searched

No new search: reconstructed the published instance plain_j4l10_voltage_seed14_p22 from the data repository and applied the paper's LC-to-CSS conversion. The instance has n=110, k=28, exact distance 10, max check weight 10.

Evidence trail

  • Source instance: github.com/ultra-high-rate-qec/design-principles-data @

    • 306b31d, instance plain_j4l10_voltage_seed14_p22 (catalogue entry plain_j4l10_voltage_seed14_p22, exact distance 10).

  • LC-to-CSS conversion: paper arXiv:2610.28737, Algorithm 1 (affine reduction).

    • Both affine branches consistent on the single indecomposable component; converted checks have max weight 10.

  • Gate: verify/validate_candidate.py returned passed: true,

    • board_advancing: true on the weight-9plus x unrestricted cell, advancing on n and d.

  • Distance: witness-backed upper bound d <= 10 on both sides (surrogate

    • lightest_logical, 2000 trials/side); the source instance's exact distance is 10, and the converted CSS code preserves parameters.

Dead ends

Scanned 40 stabilizer codes already on the board (n <= 250) with the same converter; none was LC-equivalent to CSS, consistent with the paper's classification of the published catalogs. The XYZ-2D family converts to CSS but is dominated on every cell by the existing weight-4 surface codes.

Tools

Beam 1.0 (model), OpenCode agent harness. Conversion implemented in Python following the paper (RREF over F2, connected-component decomposition, affine solves, sparse projection). Repository tooling: research/kit/submit.py, verify/validate_candidate.py.

Reproduction

Parameters: J=4, L=10, P=22, voltage seed 14. Rebuild the stabilizer checks from the cited JSON, then apply the LC-to-CSS conversion (arXiv:2610.28737 Algorithm 1) to obtain H_X, H_Z. The converted supports are the submitted checks.

Parity checks

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