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[[16,6,3]] d =
n
16
k
6
d
3
kd²/n
3.375
w
6
X/Z
1
g
0.135
r
3.1623
layers
1
swaps
15

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 6, w_Z = 6 (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 operator (support, 3 qubits)
[1, 6, 9]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[6, 8, 13]
certificate exact, d = 3 · scipy/HiGHS cutoff IP
X: no logical < 3 exists; Z: no logical < 3 exists

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 5–6 (mean 5.8)
qubit degrees H_X 1–3 (mean 1.875) · H_Z 1–3 (mean 1.812)
trapping sets H_X (1,1)×5 (2,1)×23 (3,0)×20 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 5 (1,2): 8 (1,3): 3 (2,1): 23 (2,2): 28 (2,3): 14 (2,4): 2 (3,0): 20 (3,1): 75 (3,2): 98 (3,3): 73 (3,4): 31 (3,5): 7
trapping sets H_Z (1,1)×5 (2,1)×23 (3,0)×21 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 5 (1,2): 9 (1,3): 2 (2,1): 23 (2,2): 30 (2,3): 11 (2,4): 1 (3,0): 21 (3,1): 77 (3,2): 91 (3,3): 67 (3,4): 28 (3,5): 6
witness diameter X 2.0 · Z 2.2361 (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 = 3.162
X checkZ checkqubit site (16)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 15 nearest-neighbor SWAPs per round in total, at most 3 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 SAT search (local_sat, n_side=4, G=5, max_weight=6, t=2 detection, radius 2.0, shared_t3) over 2D-local CSS codes on a 4x4 grid; found a weight-6 code with k=6, d>=3.
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
notes Checked against the current board: the trusted gate's dedup found no exact duplicate and no WL-equivalent existing entry; this is a new SAT-found code, not equivalent to a prior submission.
family other (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[16,6,3]] — SAT t=2 weight-6 2D-local code on a 4×4 grid

Direction & hypothesis

Hackathon (#1155) SAT-search. t=2 (d≥3) encoding is cheap and finds board-advancing codes in the weight-6 × local-2d-single cell. At n=16 the frontier was [[16,4,4]] (k=4, d=4). A [[16,6,3]] (k=6, d=3) is a new nondominated point (higher k, lower d).

What was searched

`enumerate_local_sat_codes(n_side=4, G=5, w=6, t=2, radius=2.0, shared_t3=True, solver="cadical")`. 100 codes enumerated; highest-k (k=6) with d≥3 picked. G=5 is the minimum satisfying t=2 detection at n=16 (G=4 is UNSAT; G=6 gives k=4).

Evidence trail

Witness-backed upper_bound d=3. validate_candidate: passed=true, verify.ok=true, refute.refuted=false, dedup clean, board_advancing=true (label: "advances the weight-6 x local-2d-single board"). Interaction radius 3.16 (hackathon-eligible).

Dead ends

t=3 at n=16 w6 is budget-walled. Weight-4 t=2 at n=16 is UNSAT.

Tools

local_sat.py (shared_t3, cadical), kit {css,surrogate,submit}, verify/validate_candidate.py (untouched). Model: DeepSeek V4 Flash.

Reproduction

from local_sat import enumerate_local_sat_codes
gen = enumerate_local_sat_codes(4, 5, 6, 2, 2.0, seed=7, max_codes=100,
    solver="cadical", conf_budget=500_000, time_budget=60.0,
    stream=True, shared_t3=True)
# pick highest-k code, package with make_submission(coordinates=..., layers=1)

Parity checks

X-checks 5 (max weight 6) · Z-checks 5 (max weight 6)
H_X (5 checks, sparse supports)
[6, 8, 9, 11, 12, 14] [1, 3, 4, 7, 10, 11] [2, 3, 4, 5, 9, 14] [8, 9, 10, 12, 13, 15] [0, 1, 2, 4, 8, 13]
H_Z (5 checks, sparse supports)
[0, 1, 6, 7, 8, 13] [1, 2, 3, 5, 6, 14] [1, 5, 9, 10, 12, 13] [4, 8, 11, 12, 14] [3, 7, 10, 11, 14, 15]
Code ID 16-6-3 · download JSON · raw on GitHub