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[[3,2,1]] d =
n
3
k
2
d
1
kd²/n
0.666667
w
3
X/Z
2
g
0.667
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 2 · d_X = 1, d_Z = 2 · w_X = 3, w_Z = 0 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 1 · witness weight 1 (claimed upper_bound)
witness operator (support, 1 qubits)
[0]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[0, 1]
certificate exact, d = 1 · 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 = 1, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 0); words of weight 1 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 acyclic · H_Z acyclic (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 3 · H_Z 0
qubit degrees H_X 1 · H_Z 0
trapping sets H_X (1,1)×3 (2,0)×3 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 3 (2,0): 3 (3,1): 1
trapping sets H_Z (1,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,0): 3
witness diameter X 0.0 · Z 1.4142 (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
r = 1.414
X checkZ checkqubit site (3)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 E-PEEL q=0 Z from 4-2-2.json
model Beam 1.0 (claimed, not verified)
builds on arXiv:quant-ph/9707021
date 2026-10-24
notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. The layout is inherited from the parent and restricted to the qubits the child kept.
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

[[3,2,1]] — priced-calculus E-PEEL

Direction & hypothesis

Aimed at a thin (n,k,d) cell. Derived by a single E-PEEL move of the priced construction calculus from a parent entry, keeping only instances with a connected Tanner graph and fresh X/Z witnesses.

What was searched

Single-move E-PEEL sweep; survivors screened for X and Z logical witnesses up to weight 4 and validated with verify/validate_candidate.py.

Evidence trail

Witness-backed upper bounds: d_X <= 1, d_Z <= 2, so d <= 1.

Dead ends

Disconnected or witness-less instances were discarded.

Tools

Model: Beam 1.0. Priced construction calculus consumer (E-PEEL cell). Challenge verifier verify/qldpc_verify.py.

Reproduction

Operation: E-PEEL q=0 Z from 4-2-2.json. Witnesses as stored in the entry.

Parity checks

X-checks 1 (max weight 3) · Z-checks 0 (max weight 0)
H_X (1 checks, sparse supports)
[0, 1, 2]
H_Z (0 checks, sparse supports)
Code ID 3-2-1 · download JSON · raw on GitHub