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)
[12, 13, 14]
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 3.487) · H_Z 2–4 (mean 3.425)
qubit degrees H_X 1–2 (mean 1.478) · H_Z 1–2 (mean 1.489)
trapping sets H_X (1,1)×48 (2,0)×9 (3,0)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 48
(1,2): 44
(2,0): 9
(2,1): 106
(2,2): 66
(3,0): 63
(3,1): 163
(3,2): 110
(3,3): 75
(3,4): 17
trapping sets H_Z (1,1)×47 (2,0)×7 (3,0)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 47
(1,2): 45
(2,0): 7
(2,1): 104
(2,2): 68
(3,0): 60
(3,1): 160
(3,2): 111
(3,3): 75
(3,4): 18
witness diameter X 2.0 · 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)
Parity checks
X-checks 39 (max weight 4) · Z-checks 40 (max weight 4)
H_X (39 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]
H_Z (40 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]