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[[176,9,4]] d =
n
176
k
9
d
4
kd²/n
0.818182
w
4
X/Z
1.25
g
0.818
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1.25 · d_X = 5, d_Z = 4 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 5 · witness weight 5 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 5 qubits)
[7, 32, 59, 88, 116]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 4 qubits)
[0, 1, 2, 3]
certificate exact, d = 4 · graphlike_A (source sha256 66a15ffb) and graphlike_B (56d04349), two independent exact programs for graph-like check matrices; github.com/TimoKellerMath/quantum-error-correction @ ae52261bfdbd, board-certificates-2026-11 · level solver: a solver verdict, no artifact a third party can check without re-running the encoding
X: graphlike_A and graphlike_B: d_X = 5, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 4); words of weight 5 rechecked; Z: graphlike_A and graphlike_B: d_Z = 4, the length of a shortest homologically nontrivial cycle of the graph of H_X (no nontrivial Z-logical of weight <= 3); words of weight 4 rechecked

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.59) · H_Z 2–4 (mean 3.56)
qubit degrees H_X 1–2 (mean 1.693) · H_Z 1–2 (mean 1.699)
trapping sets H_X (1,1)×54 (2,0)×16 (3,0)×6 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 54 (1,2): 122 (2,0): 16 (2,1): 112 (2,2): 283 (3,0): 6 (3,1): 290 (3,2): 672 (3,3): 71 (3,4): 157
trapping sets H_Z (1,1)×53 (2,0)×15 (3,0)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 53 (1,2): 123 (2,0): 15 (2,1): 108 (2,2): 286 (3,0): 4 (3,1): 286 (3,2): 675 (3,3): 67 (3,4): 160
witness diameter X 4.0 · Z 3.0 (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 @vprusso · Inherited from the laid-out parent codes/177-9-5.json: the parent's coordinates with the entry of the removed qubit (parent index 0) dropped, since this peel child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04
r = 1.414
X checkZ checkqubit site (176)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 0 nearest-neighbor SWAPs per round in total, at most 0 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 @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction priced calculus E-PEEL transform of codes/177-9-5.json
model Beam 1.0 (claimed, not verified)
date 2026-11-02
family other (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

[[176,9,4]] — priced calculus E-PEEL transform of codes/177-9-5.json

Direction & hypothesis

Weight-4 unrestricted CSS cell. Transform a connected CSS parent (codes/177-9-5.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PEEL. Parent: codes/177-9-5.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/177-9-5.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/177-9-5.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 83 (max weight 4) · Z-checks 84 (max weight 4)
H_X (83 checks, sparse supports)
[1, 2] [4, 5] [6, 7] [9, 10] [11, 12] [14, 15] [16, 17] [19, 20] [21, 22] [0, 1, 25, 26] [2, 3, 27, 28] [5, 6, 30, 31] [7, 8, 32, 33] [10, 11, 35, 36] [12, 13, 37, 38] [15, 16, 40, 41] [17, 18, 42, 43] [20, 21, 45, 46] [22, 23, 47, 48] [24, 25, 49, 50] [26, 27, 51, 52] [29, 30, 55, 56] [31, 32, 57, 58] [34, 35, 61, 62] [36, 37, 63, 64] [39, 40, 67, 68] [41, 42, 69, 70] [44, 45, 73, 74] [46, 47, 75, 76] [50, 51, 79, 80] [52, 53, 81, 82] [54, 55, 83, 84] [56, 57, 85, 86] [58, 59, 87, 88] [60, 61, 89, 90] [62, 63, 91, 92] [64, 65, 93, 94] [66, 67, 95, 96] [68, 69, 97, 98] [70, 71, 99, 100] [72, 73, 101, 102] [74, 75, 103, 104] [76, 77, 105, 106] [78, 79, 107, 108] [80, 81, 109, 110] [82, 83, 111, 112] [84, 85, 113, 114] [86, 87, 115, 116] [88, 89, 117, 118] [90, 91, 119, 120] [92, 93, 121, 122] [94, 95, 123, 124] [96, 97, 125, 126] [98, 99, 127, 128] [100, 101, 129, 130] [102, 103, 131, 132] [104, 105, 133, 134] [108, 109] [110, 111, 136, 137] [112, 113, 138, 139] [114, 115, 140] [116, 117, 141, 142] [118, 119, 143, 144] [120, 121, 145] [122, 123, 146, 147] [124, 125, 148, 149] [126, 127, 150] [128, 129, 151, 152] [130, 131, 153, 154] [132, 133, 155] [134, 135] [136, 156] [137, 138, 157, 158] [139, 140, 159, 160] [141, 161] [142, 143, 162, 163] [144, 145, 164, 165] [146, 166] [147, 148, 167, 168] [149, 150, 169, 170] [151, 171] [152, 153, 172, 173] [154, 155, 174, 175]
H_Z (84 checks, sparse supports)
[0, 24, 25] [1, 2, 26, 27] [3, 28] [4, 5, 29, 30] [6, 7, 31, 32] [8, 33] [9, 10, 34, 35] [11, 12, 36, 37] [13, 38] [14, 15, 39, 40] [16, 17, 41, 42] [18, 43] [19, 20, 44, 45] [21, 22, 46, 47] [23, 48] [24, 49] [25, 26, 50, 51] [27, 28, 52, 53] [29, 54, 55] [30, 31, 56, 57] [32, 33, 58, 59] [34, 60, 61] [35, 36, 62, 63] [37, 38, 64, 65] [39, 66, 67] [40, 41, 68, 69] [42, 43, 70, 71] [44, 72, 73] [45, 46, 74, 75] [47, 48, 76, 77] [49, 50, 78, 79] [51, 52, 80, 81] [53, 54, 82, 83] [55, 56, 84, 85] [57, 58, 86, 87] [59, 60, 88, 89] [61, 62, 90, 91] [63, 64, 92, 93] [65, 66, 94, 95] [67, 68, 96, 97] [69, 70, 98, 99] [71, 72, 100, 101] [73, 74, 102, 103] [75, 76, 104, 105] [77, 106] [78, 107] [79, 80, 108, 109] [81, 82, 110, 111] [83, 84, 112, 113] [85, 86, 114, 115] [87, 88, 116, 117] [89, 90, 118, 119] [91, 92, 120, 121] [93, 94, 122, 123] [95, 96, 124, 125] [97, 98, 126, 127] [99, 100, 128, 129] [101, 102, 130, 131] [103, 104, 132, 133] [105, 106, 134, 135] [111, 112, 137, 138] [113, 114, 139, 140] [117, 118, 142, 143] [119, 120, 144, 145] [123, 124, 147, 148] [125, 126, 149, 150] [129, 130, 152, 153] [131, 132, 154, 155] [136, 137, 156, 157] [138, 139, 158, 159] [141, 142, 161, 162] [143, 144, 163, 164] [146, 147, 166, 167] [148, 149, 168, 169] [151, 152, 171, 172] [153, 154, 173, 174] [157, 158] [159, 160] [162, 163] [164, 165] [167, 168] [169, 170] [172, 173] [174, 175]
Code ID 176-9-4 · download JSON · raw on GitHub