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[[59,4,7]] d ≤
n
59
k
4
d
7
kd²/n
3.322
w
5
X/Z
1.14
g
0.0197
r
3.6056
layers
2
swaps
111

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Distance

X/Z asymmetry 1.14 · d_X ≤ 8, d_Z ≤ 7 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[11, 18, 22, 25, 35, 46, 54, 58]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[3, 5, 15, 25, 40, 43, 46]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 4–5 (mean 4.9)
qubit degrees H_X 1–3 (mean 2.373) · H_Z 2–3 (mean 2.492)
trapping sets H_X (1,1)×2 (2,1)×2 (3,1)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 2 (1,2): 33 (1,3): 24 (2,1): 2 (2,2): 56 (2,3): 162 (2,4): 60 (3,1): 2 (3,2): 123 (3,3): 644 (3,4): 737 (3,5): 248 (3,6): 126 (3,7): 16
trapping sets H_Z (1,2)×30 (2,2)×30 (3,2)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 30 (1,3): 29 (2,2): 30 (2,3): 174 (2,4): 84 (3,2): 30 (3,3): 558 (3,4): 924 (3,5): 303 (3,6): 168 (3,7): 27
witness diameter X 5.099 · Z 3.1623 (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 = 3.606
X checkZ checkqubit site (31)2 qubits stacked (2 layers)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 111 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 Check deletion + degree-one r=1 graft of the [[60,4,8]] generalized-bicycle code
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-17
notes Checked against the board: no exact-duplicate or WL-equivalent entry (trusted gate).
family generalized bicycle (a tag, not a ranking)
locality 2D-local bilayer (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

[[59,4,7]] — check-deletion + r=1 graft of the [[60,4,8]] generalized-bicycle code

Direction & hypothesis

Issue #1155 hackathon, 2D-local priority. Composition hypothesis: deleting a check (raising k) can create a degree-one qubit that a subsequent r=1 graft removes, producing a (n−1, k+Δk, d') point neither move reaches alone.

What was searched

Source: [[60,4,8]] (weight-5 two-block group-algebra code on MC(3,10,2), two-layer 2D-local layout, family generalized-bicycle). Move: delete one check, then one degree-one r=1 graft. Original column identities tracked so the source coordinates subset honestly; layers preserved.

Evidence trail

Trusted gate (verify/validate_candidate, refute=True): passed=true, board_advancing=true in the weight-6 × local-2d-bilayer cell, no exact duplicate / WL-equivalent board entry, not refuted. Fresh witness search at submission: d ≤ 7, upper bound. Not an exact certificate; literature novelty unverified.

Dead ends

Two high-k single-layer leads ([[453,8,18]], [[452,8,17]]) were refuted at the gate and excluded. Plain deletion-control and incident-deletion-control moves produced no board-advancing survivors on their own.

Tools

Model: DeepSeek V4 Flash 0731 (Zed agent). Repo tooling: research/kit (make_submission, css GF(2) exports), verify/validate_candidate (trusted gate), verify/qldpc_verify (structural screening + board reports). Local compute only.

Reproduction

Rebuild (H_X, H_Z) from the source [[60,4,8]] two-block group-algebra matrix: 1. Delete one check row (the deleted row is recorded in the submission's provenance trail; the move raises k by the rank loss). 2. Graft a degree-one qubit: pick a column q that participates in exactly one stabilizer of some Pauli type, remove that column and its unique same-side row, and truncate opposite-type rows containing q (commutation is preserved by construction). 3. Subset the source's 2D coordinates to the surviving columns (original identities, not guessed) and keep the two-layer layout.

The method is the r=1 lattice grafting of arXiv:2504.08887 Sec. III E combined with check deletion; the exact deleted row/column identities are deterministic from the source and the move order described above. The packaged JSON carries embedded witnesses and seed for the distance claim.

Parity checks

X-checks 28 (max weight 5) · Z-checks 30 (max weight 5)
H_X (28 checks, sparse supports)
[0, 10, 22, 30, 37] [1, 11, 23, 31, 38] [2, 12, 24, 32, 39] [3, 13, 25, 30, 33] [4, 14, 26, 31, 34] [5, 15, 27, 32, 35] [6, 16, 28, 33, 36] [7, 17, 29, 34, 37] [8, 18, 20, 35, 38] [9, 19, 21, 36, 39] [2, 10, 20, 40, 46] [4, 12, 22, 41, 48] [5, 13, 23, 40, 42] [7, 15, 25, 41, 44] [8, 16, 26, 42, 45] [9, 17, 27, 43, 46] [0, 18, 28, 44, 47] [1, 19, 29, 45, 48] [0, 12, 20, 49, 56] [1, 13, 21, 50, 57] [2, 14, 22, 51, 58] [3, 15, 23, 49, 52] [4, 16, 24, 50, 53] [5, 17, 25, 51, 54] [6, 18, 26, 52, 55] [7, 19, 27, 53, 56] [8, 10, 28, 54, 57] [9, 11, 29, 55, 58]
H_Z (30 checks, sparse supports)
[0, 3, 30, 47, 49] [1, 4, 31, 48, 50] [2, 5, 32, 40, 51] [3, 6, 33, 52] [4, 7, 34, 41, 53] [5, 8, 35, 42, 54] [6, 9, 36, 43, 55] [0, 7, 37, 44, 56] [1, 8, 38, 45, 57] [2, 9, 39, 46, 58] [10, 13, 30, 40, 57] [11, 14, 31, 58] [12, 15, 32, 41, 49] [13, 16, 33, 42, 50] [14, 17, 34, 43, 51] [15, 18, 35, 44, 52] [16, 19, 36, 45, 53] [10, 17, 37, 46, 54] [11, 18, 38, 47, 55] [12, 19, 39, 48, 56] [20, 23, 38, 40, 49] [21, 24, 39, 50] [22, 25, 30, 41, 51] [23, 26, 31, 42, 52] [24, 27, 32, 43, 53] [25, 28, 33, 44, 54] [26, 29, 34, 45, 55] [20, 27, 35, 46, 56] [21, 28, 36, 47, 57] [22, 29, 37, 48, 58]
Code ID 59-4-7 · download JSON · raw on GitHub