[[77,27,5]] · QEC Challenge

[[77,27,5]] d ≤

kd²/n

8.76623

X/Z

1.2

Share this result

Distance

X/Z asymmetry 1.2 · d_X ≤ 6, d_Z ≤ 5 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 6 · witness weight 6 (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, 6 qubits)

[6, 32, 57, 58, 63, 70]

d_Z 5 · witness weight 5 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 5 qubits)

[17, 36, 59, 60, 63]

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 10 · H_Z 9–10 (mean 9.846)

qubit degrees H_X 1–5 (mean 3.117) · H_Z 1–5 (mean 3.325)

trapping sets H_X (1,1)×3 (2,1)×2 (3,1)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 3 (1,2): 12 (1,3): 40 (1,4): 17 (1,5): 5 (2,1): 2 (2,2): 35 (2,3): 157 (2,4): 324 (2,5): 274 (2,6): 154 (2,7): 45 (2,8): 6 (3,1): 16 (3,2): 156 (3,3): 753 (3,4): 2219 (3,5): 4010 (3,6): 4569 (3,7): 3573 (3,8): 1903 (3,9): 698 (3,10): 178 (3,11): 46 (3,12): 8

trapping sets H_Z (1,1)×1 (2,1)×2 (3,1)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 1 (1,2): 17 (1,3): 25 (1,4): 24 (1,5): 10 (2,1): 2 (2,2): 25 (2,3): 116 (2,4): 215 (2,5): 319 (2,6): 227 (2,7): 108 (2,8): 24 (3,1): 9 (3,2): 85 (3,3): 451 (3,4): 1430 (3,5): 3080 (3,6): 4323 (3,7): 4495 (3,8): 3346 (3,9): 1713 (3,10): 627 (3,11): 147 (3,12): 27 (3,13): 2

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PEEL transform of codes/78-27-5.json

model Beam 1.0 (claimed, not verified)

date 2026-11-02

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

[[77,27,5]] — priced calculus E-PEEL transform of codes/78-27-5.json

Direction & hypothesis

Weight-10 unrestricted CSS cell. Transform a connected CSS parent (codes/78-27-5.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

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/78-27-5.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/78-27-5.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/78-27-5.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 24 (max weight 10) · Z-checks 26 (max weight 10)

H_X (24 checks, sparse supports)

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

H_Z (26 checks, sparse supports)

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

Code ID 77-27-5 · download JSON · raw on GitHub