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[[167,8,4]] d =
n
167
k
8
d
4
kd²/n
0.766467
w
4
X/Z
1.25
g
0.766
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)
[96, 113, 128, 142, 157]
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, 16, 32, 33]
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 3.57) · H_Z 2–4 (mean 3.538)
qubit degrees H_X 1–2 (mean 1.689) · H_Z 1–2 (mean 1.695)
trapping sets H_X (1,1)×52 (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): 52 (1,2): 115 (2,0): 16 (2,1): 110 (2,2): 261 (3,0): 6 (3,1): 280 (3,2): 607 (3,3): 71 (3,4): 141
trapping sets H_Z (1,1)×51 (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): 51 (1,2): 116 (2,0): 15 (2,1): 106 (2,2): 264 (3,0): 4 (3,1): 276 (3,2): 610 (3,3): 67 (3,4): 144
witness diameter X 4.4721 · Z 3.6056 (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/168-8-5.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 (167)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/168-8-5.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

[[167,8,4]] — priced calculus E-PEEL transform of codes/168-8-5.json

Direction & hypothesis

Weight-4 unrestricted CSS cell. Transform a connected CSS parent (codes/168-8-5.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/168-8-5.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/168-8-5.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/168-8-5.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

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