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 · 2026-11-01
witness operator (support, 5 qubits)
[12, 13, 14, 15, 16]
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 · 2026-11-01
witness operator (support, 4 qubits)
[4, 9, 14, 21]
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 3.552) · H_Z 2–4 (mean 3.467)
qubit degrees H_X 1–2 (mean 1.661) · H_Z 1–2 (mean 1.677)
trapping sets H_X (1,1)×21 (2,0)×7 (3,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 21
(1,2): 41
(2,0): 7
(2,1): 43
(2,2): 91
(3,0): 3
(3,1): 107
(3,2): 210
(3,3): 26
(3,4): 49
trapping sets H_Z (1,1)×20 (2,0)×6 (3,0)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 20
(1,2): 42
(2,0): 6
(2,1): 39
(2,2): 94
(3,0): 1
(3,1): 103
(3,2): 213
(3,3): 22
(3,4): 52
witness diameter X 4.1231 · Z 3.0 (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)
Construction & provenance
provenance submitted through the challenge
novelty novelty not audited
construction priced calculus E-PEEL transform of codes/63-3-5.json
model Beam 1.0 (claimed, not verified)
date 2026-11-01
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
[[62,3,4]] — priced calculus E-PEEL transform of 63-3-5.json
Direction & hypothesis
Weight-4 unrestricted CSS cell. Transform a connected CSS parent (63-3-5.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
Single-step and depth-2 deletion (C-DELETE-CHECK), peel (E-PEEL), and puncture (E-PUNCTURE) moves enumerated with machine.planner.plan from github.com/MathysRennela/code-calculus-dev. Parent: codes/63-3-5.json. Rule used for this survivor: E-PEEL. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side (research/kit/surrogate.lightest_logical, seeds 0/1), 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. The gate's random-seed refutation found no lighter logical. Parent entry: codes/63-3-5.json.
Dead ends
Other single-step moves from the same parents were discarded as disconnected, bare-qubit children, dominated candidates, or refuted by the gate.
Tools
Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer. Approx compute: CPU only.
Reproduction
Rebuild H_X/H_Z from codes/63-3-5.json by applying the E-PEEL move of the priced calculus as recorded above; re-run the witness search at the stated budget.
Parity checks
X-checks 29 (max weight 4) · Z-checks 30 (max weight 4)
H_X (29 checks, sparse supports)
[0, 1, 5, 6]
[2, 3, 7, 8]
[4, 5, 9, 10]
[6, 7, 11, 12]
[8, 13]
[9, 14]
[10, 11, 15, 16]
[12, 13, 17, 18]
[19, 20]
[14, 15, 21, 22]
[16, 17, 23, 24]
[18, 19, 25, 26]
[24, 25, 29, 30]
[26, 27, 31, 32]
[28, 29, 35, 36]
[30, 31, 37, 38]
[33, 40]
[34, 35, 41, 42]
[36, 37, 43, 44]
[38, 39, 45, 46]
[40, 41, 47, 48]
[42, 43, 49, 50]
[44, 45, 51]
[47, 52]
[48, 49, 53, 54]
[50, 51, 55, 56]
[52, 53, 57, 58]
[54, 55, 59, 60]
[56, 61]
H_Z (30 checks, sparse supports)
[0, 1]
[2, 3]
[0, 4, 5]
[1, 2, 6, 7]
[5, 6, 10, 11]
[7, 8, 12, 13]
[9, 10, 14, 15]
[11, 12, 16, 17]
[15, 16, 22, 23]
[17, 18, 24, 25]
[19, 20, 26, 27]
[21, 22]
[23, 24, 28, 29]
[25, 26, 30, 31]
[27, 32]
[28, 34, 35]
[29, 30, 36, 37]
[31, 32, 38, 39]
[33, 34, 40, 41]
[35, 36, 42, 43]
[37, 38, 44, 45]
[39, 46]
[41, 42, 48, 49]
[43, 44, 50, 51]
[47, 48, 52, 53]
[49, 50, 54, 55]
[53, 54, 58, 59]
[55, 56, 60, 61]
[57, 58]
[59, 60]