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[[190,27,2]] d =
n
190
k
27
d
2
kd²/n
0.568421
w
4
X/Z
1.5
g
0.568
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1.5 · d_X = 3, d_Z = 2 · 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 3 · witness weight 3 (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, 3 qubits)
[172, 173, 174]
d_Z 2 · witness weight 2 (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, 2 qubits)
[2, 5]
certificate exact, d = 2 · 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 = 3, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 2); words of weight 3 rechecked; Z: graphlike_A and graphlike_B: d_Z = 2, the length of a shortest homologically nontrivial cycle of the graph of H_X (no nontrivial Z-logical of weight <= 1); words of weight 2 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 3.494) · H_Z 2–4 (mean 3.463)
qubit degrees H_X 1–2 (mean 1.489) · H_Z 1–2 (mean 1.495)
trapping sets H_X (1,1)×97 (2,0)×16 (3,0)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 97 (1,2): 93 (2,0): 16 (2,1): 218 (2,2): 143 (3,0): 126 (3,1): 345 (3,2): 236 (3,3): 159 (3,4): 38
trapping sets H_Z (1,1)×96 (2,0)×14 (3,0)×123 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 96 (1,2): 94 (2,0): 14 (2,1): 216 (2,2): 145 (3,0): 123 (3,1): 342 (3,2): 237 (3,3): 159 (3,4): 39
witness diameter X 2.2361 · 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 @vprusso · Inherited from the laid-out parent codes/191-27-3.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 (190)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/191-27-3.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

[[190,27,2]] — priced calculus E-PEEL transform of codes/191-27-3.json

Direction & hypothesis

Weight-4 unrestricted CSS cell. Transform a connected CSS parent (codes/191-27-3.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/191-27-3.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/191-27-3.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/191-27-3.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 81 (max weight 4) · Z-checks 82 (max weight 4)
H_X (81 checks, sparse supports)
[0, 1, 3, 4] [2, 3, 5, 6] [4, 7, 8] [6, 7, 9, 10] [10, 11, 14, 15] [12, 16] [13, 14, 17, 18] [16, 17, 19, 20] [18, 21, 22] [20, 21, 23, 24] [24, 25, 28, 29] [26, 30] [27, 28, 31, 32] [30, 31, 33, 34] [32, 35, 36] [34, 35, 37, 38] [38, 39, 42, 43] [40, 44] [41, 42, 45, 46] [44, 45, 47, 48] [46, 49, 50] [48, 49, 51, 52] [52, 53, 56, 57] [54, 58] [55, 56, 59, 60] [58, 59, 61, 62] [60, 63, 64] [62, 63, 65, 66] [66, 67, 70, 71] [68, 72] [69, 70, 73, 74] [72, 73, 75, 76] [74, 77, 78] [76, 77, 79, 80] [80, 81, 84, 85] [82, 86] [83, 84, 87, 88] [86, 87, 89, 90] [88, 91, 92] [90, 91, 93, 94] [94, 95, 98, 99] [96, 100] [97, 98, 101, 102] [100, 101, 103, 104] [102, 105, 106] [104, 105, 107, 108] [108, 109, 112, 113] [110, 114] [111, 112, 115, 116] [114, 115, 117, 118] [116, 119, 120] [118, 119, 121, 122] [122, 123, 126, 127] [124, 128] [125, 126, 129, 130] [128, 129, 131, 132] [130, 133, 134] [132, 133, 135, 136] [136, 137, 140, 141] [138, 142] [139, 140, 143, 144] [142, 143, 145, 146] [144, 147, 148] [146, 147, 149, 150] [150, 151, 154, 155] [152, 156] [153, 154, 157, 158] [156, 157, 159, 160] [158, 161, 162] [160, 161, 163, 164] [164, 165, 168, 169] [166, 170] [167, 168, 171, 172] [170, 171, 173, 174] [172, 175, 176] [174, 175, 177, 178] [178, 179, 182, 183] [180, 184] [181, 182, 185, 186] [184, 185, 187, 188] [186, 189]
H_Z (82 checks, sparse supports)
[0, 1] [0, 2, 3] [3, 4, 6, 7] [5, 6, 9] [7, 8, 10, 11] [9, 10, 13, 14] [11, 15] [12, 13, 16, 17] [17, 18, 20, 21] [19, 20, 23] [21, 22, 24, 25] [23, 24, 27, 28] [25, 29] [26, 27, 30, 31] [31, 32, 34, 35] [33, 34, 37] [35, 36, 38, 39] [37, 38, 41, 42] [39, 43] [40, 41, 44, 45] [45, 46, 48, 49] [47, 48, 51] [49, 50, 52, 53] [51, 52, 55, 56] [53, 57] [54, 55, 58, 59] [59, 60, 62, 63] [61, 62, 65] [63, 64, 66, 67] [65, 66, 69, 70] [67, 71] [68, 69, 72, 73] [73, 74, 76, 77] [75, 76, 79] [77, 78, 80, 81] [79, 80, 83, 84] [81, 85] [82, 83, 86, 87] [87, 88, 90, 91] [89, 90, 93] [91, 92, 94, 95] [93, 94, 97, 98] [95, 99] [96, 97, 100, 101] [101, 102, 104, 105] [103, 104, 107] [105, 106, 108, 109] [107, 108, 111, 112] [109, 113] [110, 111, 114, 115] [115, 116, 118, 119] [117, 118, 121] [119, 120, 122, 123] [121, 122, 125, 126] [123, 127] [124, 125, 128, 129] [129, 130, 132, 133] [131, 132, 135] [133, 134, 136, 137] [135, 136, 139, 140] [137, 141] [138, 139, 142, 143] [143, 144, 146, 147] [145, 146, 149] [147, 148, 150, 151] [149, 150, 153, 154] [151, 155] [152, 153, 156, 157] [157, 158, 160, 161] [159, 160, 163] [161, 162, 164, 165] [163, 164, 167, 168] [165, 169] [166, 167, 170, 171] [171, 172, 174, 175] [173, 174, 177] [175, 176, 178, 179] [177, 178, 181, 182] [179, 183] [180, 181, 184, 185] [185, 186, 188, 189] [187, 188]
Code ID 190-27-2 · download JSON · raw on GitHub