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[[138,7,4]] d =
n
138
k
7
d
4
kd²/n
0.811594
w
4
X/Z
1.25
g
0.812
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)
[0, 19, 39, 63, 86]
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, 23]
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 3.585) · H_Z 2–4 (mean 3.545)
qubit degrees H_X 1–2 (mean 1.688) · H_Z 1–2 (mean 1.696)
trapping sets H_X (1,1)×43 (2,0)×13 (3,0)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 43 (1,2): 95 (2,0): 13 (2,1): 89 (2,2): 219 (3,0): 5 (3,1): 229 (3,2): 518 (3,3): 56 (3,4): 121
trapping sets H_Z (1,1)×42 (2,0)×12 (3,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 42 (1,2): 96 (2,0): 12 (2,1): 85 (2,2): 222 (3,0): 3 (3,1): 225 (3,2): 521 (3,3): 52 (3,4): 124
witness diameter X 4.0 · Z 3.1623 (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/139-7-5.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 (138)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/139-7-5.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

[[138,7,4]] — priced calculus E-PEEL transform of codes/139-7-5.json

Direction & hypothesis

Weight-4 unrestricted CSS cell. Transform a connected CSS parent (codes/139-7-5.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/139-7-5.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/139-7-5.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/139-7-5.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

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