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[[169,10,3]] d ≤
n
169
k
10
d
3
kd²/n
0.533
w
4
g
0.532
r
1.4142
layers
1

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Distance

d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[59, 73, 87]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[134, 147, 159]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 1.414
X checkZ checkqubit site (169)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

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Holey rotated surface code on integer grid L=13, period-3 hole sublattice: X plaquettes on even-parity cells, Z on odd-parity cells, X-holes at (3a,3b) even parity, Z-holes at (3a,3b) odd parity, margin 3 from boundaries (smallest size of the family generalizing codes/625-50-3.json); standard rotated-surface boundary pairs (X chains vertical, Z chains horizontal).
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-08-09
family topological (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

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

[[169,10,3]] — holey rotated surface code, L=13 period-3 hole pattern

Direction & hypothesis

Target cell: weight-4 x local-2d-single, geometric efficiency g = 4kd^2/(n·rho^2·r^4). The board's best non-reference point is the holey rotated surface code [[625,50,3]] (L = 25) at g = 0.72 with r = sqrt(2), rho = 1. This code is the L = 13 instance of the same period-3 hole pattern (n = 169, k = 10, g = 0.5325): the smallest verified member of the family, confirming the mechanism scales across patch sizes (L = 13, 19, 25).

What was searched

The period-3 hole sublattice was reverse-engineered from codes/625-50-3.json and generalized in research/candidates/oppC/src/c_holey.py: X-holes at cells (3a, 3b) with (i+j) even, Z-holes at cells (3a, 3b) with (i+j) odd, margin 3 from every boundary, standard rotated-surface boundary pairs (X chains on the vertical edges, Z chains on the horizontal edges). At L = 13 the margin-3 constraint leaves a single sublattice period (3a, 3b with a, b in {1, 2, 3} → 9 X-hole positions, 9 Z-hole positions on the parity-matched sublattice), giving k = 10. L = 19 ([[361,26,3]], PR #441) and L = 25 ([[625,50,3]], board) were built with the same builder; k tracks the hole count (10 / 26 / 50).

Evidence trail

  • CSS, n = 169, k = 10, max check weight 4 (X 67 w4 + 12 w2 = 79 rows,

    • Z 68 w4 + 12 w2 = 80 rows).

  • Distance: X witness weight 3, Z witness weight 3 (RIS search at 6,000

    • trials); both verified in the opposite kernel and outside their own rowspace; refutation clean at 8,000 trials.

  • Locality: single layer (layers = 1), interaction radius sqrt(2) (integer

    • grid, unit plaquettes), within the local-2d-single cap of 4.0.

  • Final claim: witness-backed upper bound, d <= 3. Not exact-certified

    • (n = 169, k = 10 within the certification envelope in principle, but no MILP certification was run for this submission).

  • verify/validate_candidate.py → passed: true, board_advancing: true,

    • dominated_by: [] in the weight-4 x local-2d-single cell.

Dead ends

  • L = 11 (n = 121) would put the margin-3 holes at the pattern's minimum, but

    • the 3a, b in {1, 2} positions leave too few cells for d = 3 (boundaries pinch the hole lattice); L = 13 is the smallest size with a clean three-period sublattice.

  • L = 31 is over the n <= 700 cap (n = 961); L = 27 gives n = 729 > 700, so

    • L = 25 is the largest odd L under the cap.

  • The radius-audit dense-row bug (bogus r = 1.0) is documented in the

    • L = 19 note; the corrected r = sqrt(2) is used here.

Tools

Model: DeepSeek V4 Flash 0731 (matches provenance.model). Repo kit (research/kit/css.py, research/kit/submit.py), verify/validate_candidate.py as the gate, cli/qldpc.py submit for witness search and PR drafting. Compute: ~1 CPU-minute (L = 13 is the cheapest size: 6k-trials RIS at n = 169).

Reproduction

research/candidates/oppC/src/c_holey.py: holey_rotated(13); export via research/candidates/oppC/src/c_export.py (hx/hz/coords in .npz), then

uv run python cli/qldpc.py submit holey_L13.npz --authors @mathysrennela \
  --model "DeepSeek V4 Flash 0731" --family topological --layers 1 --trials 6000

Hole coordinates: X-holes at (3a, 3b) with (i+j) even in [3, L-4]^2, Z-holes at (3a, 3b) with (i+j) odd; margin 3 from every boundary.

Parity checks

X-checks 79 · Z-checks 80
H_X (79 checks, sparse supports)
[0, 1, 13, 14] [2, 3, 15, 16] [4, 5, 17, 18] [6, 7, 19, 20] [8, 9, 21, 22] [10, 11, 23, 24] [14, 15, 27, 28] [16, 17, 29, 30] [18, 19, 31, 32] [20, 21, 33, 34] [22, 23, 35, 36] [24, 25, 37, 38] [26, 27, 39, 40] [28, 29, 41, 42] [30, 31, 43, 44] [32, 33, 45, 46] [34, 35, 47, 48] [36, 37, 49, 50] [40, 41, 53, 54] [44, 45, 57, 58] [46, 47, 59, 60] [50, 51, 63, 64] [52, 53, 65, 66] [54, 55, 67, 68] [56, 57, 69, 70] [58, 59, 71, 72] [60, 61, 73, 74] [62, 63, 75, 76] [66, 67, 79, 80] [68, 69, 81, 82] [70, 71, 83, 84] [72, 73, 85, 86] [74, 75, 87, 88] [76, 77, 89, 90] [78, 79, 91, 92] [80, 81, 93, 94] [82, 83, 95, 96] [86, 87, 99, 100] [88, 89, 101, 102] [92, 93, 105, 106] [94, 95, 107, 108] [96, 97, 109, 110] [98, 99, 111, 112] [100, 101, 113, 114] [102, 103, 115, 116] [104, 105, 117, 118] [106, 107, 119, 120] [108, 109, 121, 122] [110, 111, 123, 124] [112, 113, 125, 126] [114, 115, 127, 128] [118, 119, 131, 132] [122, 123, 135, 136] [124, 125, 137, 138] [128, 129, 141, 142] [130, 131, 143, 144] [132, 133, 145, 146] [134, 135, 147, 148] [136, 137, 149, 150] [138, 139, 151, 152] [140, 141, 153, 154] [144, 145, 157, 158] [146, 147, 159, 160] [148, 149, 161, 162] [150, 151, 163, 164] [152, 153, 165, 166] [154, 155, 167, 168] [13, 26] [39, 52] [65, 78] [91, 104] [117, 130] [143, 156] [12, 25] [38, 51] [64, 77] [90, 103] [116, 129] [142, 155]
H_Z (80 checks, sparse supports)
[1, 2, 14, 15] [3, 4, 16, 17] [5, 6, 18, 19] [7, 8, 20, 21] [9, 10, 22, 23] [11, 12, 24, 25] [13, 14, 26, 27] [15, 16, 28, 29] [17, 18, 30, 31] [19, 20, 32, 33] [21, 22, 34, 35] [23, 24, 36, 37] [27, 28, 40, 41] [29, 30, 42, 43] [31, 32, 44, 45] [33, 34, 46, 47] [35, 36, 48, 49] [37, 38, 50, 51] [39, 40, 52, 53] [41, 42, 54, 55] [43, 44, 56, 57] [47, 48, 60, 61] [49, 50, 62, 63] [53, 54, 66, 67] [55, 56, 68, 69] [57, 58, 70, 71] [59, 60, 72, 73] [61, 62, 74, 75] [63, 64, 76, 77] [65, 66, 78, 79] [67, 68, 80, 81] [69, 70, 82, 83] [71, 72, 84, 85] [73, 74, 86, 87] [75, 76, 88, 89] [79, 80, 92, 93] [83, 84, 96, 97] [85, 86, 98, 99] [89, 90, 102, 103] [91, 92, 104, 105] [93, 94, 106, 107] [95, 96, 108, 109] [97, 98, 110, 111] [99, 100, 112, 113] [101, 102, 114, 115] [105, 106, 118, 119] [107, 108, 120, 121] [109, 110, 122, 123] [111, 112, 124, 125] [113, 114, 126, 127] [115, 116, 128, 129] [117, 118, 130, 131] [119, 120, 132, 133] [121, 122, 134, 135] [125, 126, 138, 139] [127, 128, 140, 141] [131, 132, 144, 145] [133, 134, 146, 147] [135, 136, 148, 149] [137, 138, 150, 151] [139, 140, 152, 153] [141, 142, 154, 155] [143, 144, 156, 157] [145, 146, 158, 159] [147, 148, 160, 161] [149, 150, 162, 163] [151, 152, 164, 165] [153, 154, 166, 167] [0, 1] [2, 3] [4, 5] [6, 7] [8, 9] [10, 11] [157, 158] [159, 160] [161, 162] [163, 164] [165, 166] [167, 168]
raw submission JSON