[[78,44,3]] · QEC Challenge

[[78,44,3]] d ≤

kd²/n

5.07692

X/Z

1.67

Share this result

Distance

X/Z asymmetry 1.67 · d_X ≤ 5, d_Z ≤ 3 · w_X = 18, w_Z = 18 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 5 · witness weight 5 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-01

witness operator (support, 5 qubits)

[7, 38, 47, 63, 69]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-01

witness operator (support, 3 qubits)

[18, 33, 59]

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 13–18 (mean 15.75) · H_Z 11–18 (mean 16.056)

qubit degrees H_X 1–7 (mean 3.231) · H_Z 1–7 (mean 3.705)

trapping sets H_X (1,1)×4 (2,1)×20 (3,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 4 (1,2): 15 (1,3): 30 (1,4): 19 (1,5): 9 (1,7): 1 (2,1): 20 (2,2): 101 (2,3): 223 (2,4): 397 (2,5): 383 (2,6): 251 (2,7): 96 (2,8): 29 (2,9): 10 (2,10): 1 (3,0): 3 (3,1): 145 (3,2): 725 (3,3): 2479 (3,4): 5183 (3,5): 7743 (3,6): 8187 (3,7): 6060 (3,8): 3343 (3,9): 1385 (3,10): 431 (3,11): 123 (3,12): 20 (3,13): 2

trapping sets H_Z (1,1)×3 (2,1)×5 (3,1)×59 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 3 (1,2): 9 (1,3): 27 (1,4): 18 (1,5): 14 (1,6): 4 (1,7): 3 (2,1): 5 (2,2): 50 (2,3): 155 (2,4): 299 (2,5): 362 (2,6): 356 (2,7): 238 (2,8): 128 (2,9): 51 (2,10): 11 (2,11): 2 (3,1): 59 (3,2): 324 (3,3): 1243 (3,4): 3292 (3,5): 5956 (3,6): 8144 (3,7): 8429 (3,8): 6713 (3,9): 4128 (3,10): 2078 (3,11): 759 (3,12): 237 (3,13): 55 (3,14): 6 (3,15): 1

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PEEL transform of codes/80-44-5.json

model Beam 1.0 (claimed, not verified)

date 2026-11-01

family other (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight > 8 (computed)

How this code was found

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

[[78,44,3]] — priced calculus E-PEEL transform of 80-44-5.json

Direction & hypothesis

Weight-18 unrestricted CSS cell. Transform a connected CSS parent (80-44-5.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

Single-step and depth-2 deletion (C-DELETE-CHECK), peel (E-PEEL), and puncture (E-PUNCTURE) moves enumerated with machine.planner.plan from github.com/MathysRennela/code-calculus-dev. Parent: codes/80-44-5.json. Rule used for this survivor: E-PEEL. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side (research/kit/surrogate.lightest_logical, seeds 0/1), 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. The gate's random-seed refutation found no lighter logical. Parent entry: codes/80-44-5.json.

Dead ends

Other single-step moves from the same parents were discarded as disconnected, bare-qubit children, dominated candidates, or refuted by the gate.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/80-44-5.json by applying the E-PEEL move of the priced calculus as recorded above; re-run the witness search at the stated budget.

Parity checks

X-checks 16 (max weight 18) · Z-checks 18 (max weight 18)

H_X (16 checks, sparse supports)

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

H_Z (18 checks, sparse supports)

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

Code ID 78-44-3 · download JSON · raw on GitHub