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)
[80, 91, 92, 93, 94]
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, 18]
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 3.574) · H_Z 2–4 (mean 3.521)
qubit degrees H_X 1–2 (mean 1.68) · H_Z 1–2 (mean 1.69)
trapping sets H_X (1,1)×32 (2,0)×10 (3,0)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 32
(1,2): 68
(2,0): 10
(2,1): 66
(2,2): 155
(3,0): 4
(3,1): 168
(3,2): 364
(3,3): 41
(3,4): 85
trapping sets H_Z (1,1)×31 (2,0)×9 (3,0)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 31
(1,2): 69
(2,0): 9
(2,1): 62
(2,2): 158
(3,0): 2
(3,1): 164
(3,2): 367
(3,3): 37
(3,4): 88
witness diameter X 4.1231 · 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)
Parity checks
X-checks 47 (max weight 4) · Z-checks 48 (max weight 4)
H_X (47 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, 35, 36]
[21, 22, 37, 38]
[24, 25, 41, 42]
[26, 27, 43, 44]
[30, 31, 47, 48]
[32, 33, 49, 50]
[34, 35, 51, 52]
[36, 37, 53, 54]
[38, 39, 55, 56]
[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]
[60, 61, 77, 78]
[64, 65]
[66, 67, 80, 81]
[68, 69, 82, 83]
[70, 71, 84]
[72, 73, 85, 86]
[74, 75, 87, 88]
[76, 77, 89]
[78, 79]
[80, 90]
[81, 82, 91, 92]
[83, 84, 93, 94]
[85, 95]
[86, 87, 96, 97]
[88, 89, 98, 99]
H_Z (48 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, 35]
[20, 21, 36, 37]
[22, 23, 38, 39]
[24, 40, 41]
[25, 26, 42, 43]
[27, 28, 44, 45]
[29, 30, 46, 47]
[31, 32, 48, 49]
[33, 34, 50, 51]
[35, 36, 52, 53]
[37, 38, 54, 55]
[39, 40, 56, 57]
[41, 42, 58, 59]
[43, 44, 60, 61]
[45, 62]
[46, 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]
[61, 62, 78, 79]
[67, 68, 81, 82]
[69, 70, 83, 84]
[73, 74, 86, 87]
[75, 76, 88, 89]
[80, 81, 90, 91]
[82, 83, 92, 93]
[85, 86, 95, 96]
[87, 88, 97, 98]
[91, 92]
[93, 94]
[96, 97]
[98, 99]