← back to the board
[[36,12,3]] d =
n
36
k
12
d
3
kd²/n
3.0
w
6
X/Z
1
g
0.0469
r
4.0
layers
1
swaps
50

Share this result

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)
[24, 27, 31]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[6, 13, 24]
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 4–6 (mean 5.667) · H_Z 4–6 (mean 5.667)
qubit degrees H_X 1–4 (mean 1.889) · H_Z 1–4 (mean 1.889)
trapping sets H_X (1,1)×12 (2,1)×43 (3,0)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 12 (1,2): 17 (1,3): 6 (1,4): 1 (2,1): 43 (2,2): 58 (2,3): 33 (2,4): 10 (2,5): 3 (3,0): 30 (3,1): 120 (3,2): 207 (3,3): 223 (3,4): 136 (3,5): 53 (3,6): 11 (3,7): 3 (3,8): 1
trapping sets H_Z (1,1)×10 (2,1)×38 (3,0)×26 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 10 (1,2): 21 (1,3): 4 (1,4): 1 (2,1): 38 (2,2): 69 (2,3): 32 (2,4): 12 (2,5): 2 (3,0): 26 (3,1): 121 (3,2): 232 (3,3): 247 (3,4): 156 (3,5): 53 (3,6): 15 (3,7): 3
witness diameter X 3.0 · 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)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 4
X checkZ checkqubit site (36)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 50 nearest-neighbor SWAPs per round in total, at most 5 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=6, G=12, max_weight=6, t=2 detection, radius 2.0, shared_t3) over 2D-local CSS codes on a 6x6 grid; found a weight-6 code with k=12, 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

[[36,12,3]] — SAT t=2 weight-6 2D-local code on a 6×6 grid

Direction & hypothesis

Hackathon (#1155) SAT-search. Prior SAT campaigns mined t=3 (d≥4) at small grids and walled. Hypothesis: t=2 (d≥3) is a much cheaper encoding that still finds board-advancing codes in the sparse weight-6 × local-2d-single cell, where the frontier at n=36 was [[36,6,4]] (k=6, d=4). A higher-k d=3 point would be a new nondominated frontier record.

What was searched

local_sat.py `enumerate_local_sat_codes(n_side=6, G=12, w=6, t=2, radius=2.0, shared_t3=True, solver="cadical")`. 120 codes enumerated; the highest-k (k=12) code with d≥3 was picked. G=12 is the minimum that satisfies t=2 detection at n=36 (G=11 is budget-walled; G=13+ drops k).

Evidence trail

Witness-backed upper_bound d=3 (weight-3 logicals on both X and Z sides). 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 4.00 (hackathon-eligible).

Dead ends

t=3 (d≥4) at n=36 w6 is budget-walled in-session (needs overnight). Weight-8 t=3 at n=36 (radius 2.0) is empty; radius 4.0 is budget-walled. n=49 (7×7) w6/t2 is budget-walled.

Tools

local_sat.py (shared_t3, cadical, per-solve budgets), 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(6, 12, 6, 2, 2.0, seed=7, max_codes=120,
    solver="cadical", conf_budget=1_000_000, time_budget=90.0,
    stream=True, shared_t3=True)
# pick highest-k code, package with make_submission(coordinates=..., layers=1)

Parity checks

X-checks 12 (max weight 6) · Z-checks 12 (max weight 6)
H_X (12 checks, sparse supports)
[14, 19, 21, 25, 28, 33] [3, 4, 5, 16, 17, 22] [20, 26, 30, 31, 33, 34] [0, 6, 7, 14, 19, 24] [22, 29, 32, 35] [16, 23, 28, 33, 34, 35] [2, 7, 9, 15, 19, 26] [6, 12, 18, 24, 25] [5, 8, 10, 11, 15, 17] [6, 13, 18, 25, 30] [8, 14, 20, 25, 27, 32] [1, 2, 4, 5, 8, 10]
H_Z (12 checks, sparse supports)
[0, 6, 12, 13] [2, 8, 9, 15, 20, 26] [0, 1, 3, 4, 7, 9] [1, 6, 8, 15, 19, 25] [16, 21, 22, 33, 34, 35] [4, 10, 17, 21, 23, 28] [7, 13, 18, 24, 26, 31] [19, 24, 25, 26, 27, 30] [11, 16, 17, 29, 35] [9, 14, 19, 20, 28, 33] [27, 31, 32, 34, 35] [3, 5, 9, 10, 11, 15]
Code ID 36-12-3 · download JSON · raw on GitHub