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[[151,6,8]] d =
n
151
k
6
d
8
kd²/n
2.543
w
5
X/Z
1.38
g
0.002
r
6.0
layers
2
swaps
692

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Distance

X/Z asymmetry 1.38 · d_X = 11, d_Z = 8 · 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 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[14, 21, 28, 55, 62, 69, 76, 80, 84, 122, 138]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[26, 46, 50, 66, 86, 93, 100, 121]
certificate exact, d = 8 · scipy/HiGHS MILP
X: no logical < 11 exists; Z: no logical < 8 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 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 4.883)
qubit degrees H_X 1–3 (mean 2.417) · H_Z 2–3 (mean 2.49)
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): 84 (1,3): 65 (2,1): 2 (2,2): 125 (2,3): 432 (2,4): 171 (3,1): 2 (3,2): 226 (3,3): 1622 (3,4): 2070 (3,5): 735 (3,6): 360 (3,7): 49
trapping sets H_Z (1,2)×77 (2,2)×77 (3,2)×77 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 77 (1,3): 74 (2,2): 77 (2,3): 444 (2,4): 213 (3,2): 77 (3,3): 1400 (3,4): 2349 (3,5): 834 (3,6): 426 (3,7): 68
witness diameter X 6.3246 · Z 9.2195 (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 = 6
X checkZ checkqubit site (77)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 692 nearest-neighbor SWAPs per round in total, at most 9 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 + three degree-one r=1 grafts of the [[154,6,11]] 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

[[151,6,8]] — check-deletion + three r=1 grafts of the [[154,6,11]] generalized-bicycle code

Direction & hypothesis

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

What was searched

Source: [[154,6,11]] (weight-5 cyclic generalized-bicycle code on Z_77, two-layer 2D-local layout, family generalized-bicycle). Move: delete one check, then three degree-one r=1 grafts. 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 ≤ 8, upper bound. Not an exact certificate; literature novelty unverified.

Dead ends

The [[154,6,11]] family also yields [[153,6,10]] (1 graft) and [[152,6,9]] (2 grafts), submitted separately. Two high-k single-layer leads ([[453,8,18]], [[452,8,17]]) were refuted at the gate and excluded.

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 [[154,6,11]] generalized-bicycle 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 three degree-one qubits in sequence: for each, 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 73 (max weight 5) · Z-checks 77 (max weight 5)
H_X (73 checks, sparse supports)
[0, 57, 73, 77, 144] [1, 58, 74, 78, 145] [2, 59, 75, 79, 146] [3, 60, 76, 80, 147] [0, 4, 61, 81, 148] [1, 5, 62, 82, 149] [2, 6, 63, 83, 150] [3, 7, 64, 77, 84] [4, 8, 65, 78, 85] [5, 9, 66, 79, 86] [6, 10, 67, 80, 87] [7, 11, 68, 81, 88] [8, 12, 69, 82, 89] [9, 13, 70, 83, 90] [10, 14, 71, 84, 91] [11, 15, 72, 85, 92] [12, 16, 73, 86, 93] [13, 17, 74, 87, 94] [14, 18, 75, 88, 95] [15, 19, 76, 89, 96] [0, 16, 20, 90, 97] [1, 17, 21, 91, 98] [2, 18, 22, 92, 99] [3, 19, 23, 93, 100] [4, 20, 24, 94, 101] [5, 21, 25, 95, 102] [7, 23, 27, 97, 103] [8, 24, 28, 98, 104] [9, 25, 29, 99, 105] [10, 26, 30, 100, 106] [11, 27, 31, 101, 107] [12, 28, 32, 102, 108] [14, 30, 34, 103, 109] [15, 31, 35, 104, 110] [16, 32, 36, 105, 111] [17, 33, 37, 106, 112] [18, 34, 38, 107, 113] [19, 35, 39, 108, 114] [21, 37, 41, 109, 115] [22, 38, 42, 110, 116] [23, 39, 43, 111, 117] [24, 40, 44, 112, 118] [25, 41, 45, 113, 119] [26, 42, 46, 114, 120] [28, 44, 48, 115, 122] [29, 45, 49, 116, 123] [30, 46, 50, 117, 124] [31, 47, 51, 118, 125] [32, 48, 52, 119, 126] [33, 49, 53, 120, 127] [34, 50, 54, 121, 128] [35, 51, 55, 122, 129] [36, 52, 56, 123, 130] [37, 53, 57, 124, 131] [38, 54, 58, 125, 132] [39, 55, 59, 126, 133] [40, 56, 60, 127, 134] [41, 57, 61, 128, 135] [42, 58, 62, 129, 136] [43, 59, 63, 130, 137] [44, 60, 64, 131, 138] [45, 61, 65, 132, 139] [46, 62, 66, 133, 140] [47, 63, 67, 134, 141] [48, 64, 68, 135, 142] [49, 65, 69, 136, 143] [50, 66, 70, 137, 144] [51, 67, 71, 138, 145] [52, 68, 72, 139, 146] [53, 69, 73, 140, 147] [54, 70, 74, 141, 148] [55, 71, 75, 142, 149] [56, 72, 76, 143, 150]
H_Z (77 checks, sparse supports)
[0, 7, 77, 81, 97] [1, 8, 78, 82, 98] [2, 9, 79, 83, 99] [3, 10, 80, 84, 100] [4, 11, 81, 85, 101] [5, 12, 82, 86, 102] [6, 13, 83, 87] [7, 14, 84, 88, 103] [8, 15, 85, 89, 104] [9, 16, 86, 90, 105] [10, 17, 87, 91, 106] [11, 18, 88, 92, 107] [12, 19, 89, 93, 108] [13, 20, 90, 94] [14, 21, 91, 95, 109] [15, 22, 92, 96, 110] [16, 23, 93, 97, 111] [17, 24, 94, 98, 112] [18, 25, 95, 99, 113] [19, 26, 96, 100, 114] [20, 27, 97, 101] [21, 28, 98, 102, 115] [22, 29, 99, 116] [23, 30, 100, 103, 117] [24, 31, 101, 104, 118] [25, 32, 102, 105, 119] [26, 33, 106, 120] [27, 34, 103, 107, 121] [28, 35, 104, 108, 122] [29, 36, 105, 123] [30, 37, 106, 109, 124] [31, 38, 107, 110, 125] [32, 39, 108, 111, 126] [33, 40, 112, 127] [34, 41, 109, 113, 128] [35, 42, 110, 114, 129] [36, 43, 111, 130] [37, 44, 112, 115, 131] [38, 45, 113, 116, 132] [39, 46, 114, 117, 133] [40, 47, 118, 134] [41, 48, 115, 119, 135] [42, 49, 116, 120, 136] [43, 50, 117, 121, 137] [44, 51, 118, 122, 138] [45, 52, 119, 123, 139] [46, 53, 120, 124, 140] [47, 54, 121, 125, 141] [48, 55, 122, 126, 142] [49, 56, 123, 127, 143] [50, 57, 124, 128, 144] [51, 58, 125, 129, 145] [52, 59, 126, 130, 146] [53, 60, 127, 131, 147] [54, 61, 128, 132, 148] [55, 62, 129, 133, 149] [56, 63, 130, 134, 150] [57, 64, 77, 131, 135] [58, 65, 78, 132, 136] [59, 66, 79, 133, 137] [60, 67, 80, 134, 138] [61, 68, 81, 135, 139] [62, 69, 82, 136, 140] [63, 70, 83, 137, 141] [64, 71, 84, 138, 142] [65, 72, 85, 139, 143] [66, 73, 86, 140, 144] [67, 74, 87, 141, 145] [68, 75, 88, 142, 146] [69, 76, 89, 143, 147] [0, 70, 90, 144, 148] [1, 71, 91, 145, 149] [2, 72, 92, 146, 150] [3, 73, 77, 93, 147] [4, 74, 78, 94, 148] [5, 75, 79, 95, 149] [6, 76, 80, 96, 150]
Code ID 151-6-8 · download JSON · raw on GitHub