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[[77,10,2]] d =
n
77
k
10
d
2
kd²/n
0.519481
w
4
X/Z
1.5
g
0.519
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1.5 · d_X = 3, d_Z = 2 · w_X = 4, w_Z = 4 (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 found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 3 qubits)
[5, 6, 7]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 2 qubits)
[5, 11]
certificate exact, d = 2 · 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 = 3, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 2); words of weight 3 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 12 · H_Z 12 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.394) · H_Z 2–4 (mean 3.324)
qubit degrees H_X 1–2 (mean 1.455) · H_Z 1–2 (mean 1.468)
trapping sets H_X (1,1)×42 (2,0)×10 (3,0)×53 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 42 (1,2): 35 (2,0): 10 (2,1): 86 (2,2): 49 (3,0): 53 (3,1): 118 (3,2): 84 (3,3): 56 (3,4): 12
trapping sets H_Z (1,1)×41 (2,0)×7 (3,0)×51 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 41 (1,2): 36 (2,0): 7 (2,1): 86 (2,2): 50 (3,0): 51 (3,1): 114 (3,2): 85 (3,3): 57 (3,4): 12
witness diameter X 2.0 · Z 1.0 (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
layout contributed by @vprusso · Inherited from the laid-out parent codes/78-10-3.json: the parent's coordinates with the entry of the removed qubit (parent index 0) dropped, since this peel child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04
r = 1.414
X checkZ checkqubit site (77)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 priced calculus E-PEEL transform of codes/78-10-3.json
model Beam 1.0 (claimed, not verified)
date 2026-11-02
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

[[77,10,2]] — priced calculus E-PEEL transform of codes/78-10-3.json

Direction & hypothesis

Weight-4 unrestricted CSS cell. Transform a connected CSS parent (codes/78-10-3.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PEEL. Parent: codes/78-10-3.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/78-10-3.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/78-10-3.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 33 (max weight 4) · Z-checks 34 (max weight 4)
H_X (33 checks, sparse supports)
[0, 1, 6, 7] [2, 8] [3, 4, 9, 10] [5, 6, 11, 12] [7, 13, 14] [8, 9, 15, 16] [10, 17, 18] [14, 15, 21, 22] [18, 19, 24, 25] [20, 21, 28, 29] [23, 24, 32, 33] [26, 35] [27, 28, 36, 37] [29, 30, 38] [31, 32, 39, 40] [33, 34] [35, 36, 41, 42] [37, 43, 44] [38, 39, 45, 46] [40, 47, 48] [44, 45, 51, 52] [48, 49, 54, 55] [50, 51, 58, 59] [53, 54, 62, 63] [56, 65] [57, 58, 66, 67] [59, 60, 68] [61, 62, 69, 70] [63, 64] [65, 66, 71, 72] [67, 73] [68, 69, 74, 75] [70, 76]
H_Z (34 checks, sparse supports)
[0, 1] [3, 4] [0, 5, 6] [2, 3, 8, 9] [6, 7, 12, 13] [9, 10, 16, 17] [11, 12] [13, 14, 20, 21] [15, 16, 22] [17, 18, 23, 24] [19, 25] [20, 27, 28] [21, 22, 29, 30] [23, 31, 32] [24, 25, 33, 34] [26, 27, 35, 36] [30, 31, 38, 39] [36, 37, 42, 43] [39, 40, 46, 47] [41, 42] [43, 44, 50, 51] [45, 46, 52] [47, 48, 53, 54] [49, 55] [50, 57, 58] [51, 52, 59, 60] [53, 61, 62] [54, 55, 63, 64] [56, 57, 65, 66] [60, 61, 68, 69] [66, 67, 72, 73] [69, 70, 75, 76] [71, 72] [74, 75]
Code ID 77-10-2 · download JSON · raw on GitHub