[[55,14,7]] · QEC Challenge

[[55,14,7]] d ≤

kd²/n

12.4727

X/Z

1.14

Share this result

Distance

X/Z asymmetry 1.14 · d_X ≤ 7, d_Z ≤ 8 · 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 7 · witness weight 7 (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, 7 qubits)

[12, 23, 29, 31, 38, 44, 54]

d_Z 8 · witness weight 8 (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, 8 qubits)

[10, 12, 36, 39, 42, 43, 44, 47]

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 9–10 (mean 9.81) · H_Z 10

qubit degrees H_X 2–6 (mean 3.745) · H_Z 2–6 (mean 3.636)

trapping sets H_X (1,2)×9 (2,2)×9 (3,1)×6 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 9 (1,3): 14 (1,4): 17 (1,5): 12 (1,6): 3 (2,2): 9 (2,3): 37 (2,4): 120 (2,5): 196 (2,6): 183 (2,7): 129 (2,8): 71 (2,9): 21 (2,10): 1 (3,1): 6 (3,2): 28 (3,3): 175 (3,4): 612 (3,5): 1392 (3,6): 2302 (3,7): 2808 (3,8): 2619 (3,9): 1787 (3,10): 930 (3,11): 331 (3,12): 93 (3,13): 40 (3,14): 2

trapping sets H_Z (1,2)×9 (2,1)×1 (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

(1,2): 9 (1,3): 19 (1,4): 13 (1,5): 11 (1,6): 3 (2,1): 1 (2,2): 13 (2,3): 48 (2,4): 153 (2,5): 181 (2,6): 174 (2,7): 124 (2,8): 50 (2,9): 17 (2,10): 1 (3,1): 9 (3,2): 46 (3,3): 246 (3,4): 785 (3,5): 1678 (3,6): 2427 (3,7): 2848 (3,8): 2485 (3,9): 1466 (3,10): 688 (3,11): 222 (3,12): 72 (3,13): 19 (3,14): 2

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PEEL transform of codes/56-14-8.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

[[55,14,7]] — priced calculus E-PEEL transform of 56-14-8.json

Direction & hypothesis

Weight-10 unrestricted CSS cell. Transform a connected CSS parent (56-14-8.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/56-14-8.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/56-14-8.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/56-14-8.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 21 (max weight 10) · Z-checks 20 (max weight 10)

H_X (21 checks, sparse supports)

[0, 14, 22, 27, 28, 29, 33, 42, 45, 47] [1, 15, 23, 26, 30, 34, 41, 46, 48] [2, 12, 20, 25, 28, 29, 31, 40, 43, 49] [3, 13, 21, 24, 30, 32, 39, 44, 50] [3, 4, 18, 26, 31, 33, 37, 46, 49, 51] [2, 5, 19, 27, 32, 34, 38, 45, 50, 52] [1, 6, 16, 24, 31, 33, 35, 44, 47, 53] [0, 7, 17, 25, 32, 34, 36, 43, 48, 54] [2, 7, 8, 22, 28, 35, 37, 41, 50, 53] [3, 6, 9, 23, 36, 38, 42, 49, 54] [0, 5, 10, 20, 29, 35, 37, 39, 48, 51] [1, 4, 11, 21, 30, 36, 38, 40, 47, 52] [6, 11, 12, 26, 29, 31, 39, 41, 45, 54] [7, 10, 13, 27, 30, 32, 40, 42, 46, 53] [4, 9, 14, 24, 28, 33, 39, 41, 43, 52] [2, 10, 15, 16, 30, 33, 35, 43, 45, 49] [3, 11, 14, 17, 29, 34, 36, 44, 46, 50] [0, 8, 13, 18, 31, 37, 43, 45, 47] [6, 14, 19, 20, 34, 37, 39, 47, 49, 53] [7, 15, 18, 21, 33, 38, 40, 48, 50, 54] [10, 18, 23, 24, 29, 38, 41, 43, 51, 53]

H_Z (20 checks, sparse supports)

[0, 2, 8, 14, 19, 26, 28, 34, 37, 45] [0, 2, 10, 12, 17, 24, 29, 32, 35, 43] [1, 3, 11, 13, 16, 25, 30, 31, 36, 44] [2, 4, 6, 12, 18, 23, 31, 38, 41, 49] [3, 5, 7, 13, 19, 22, 32, 37, 42, 50] [0, 4, 6, 14, 16, 21, 33, 36, 39, 47] [1, 5, 7, 15, 17, 20, 34, 35, 40, 48] [6, 8, 10, 16, 22, 27, 35, 42, 45, 53] [7, 9, 11, 17, 23, 26, 36, 41, 46, 54] [4, 8, 10, 18, 20, 25, 37, 40, 43, 51] [5, 9, 11, 19, 21, 24, 38, 39, 44, 52] [3, 10, 12, 14, 20, 26, 29, 39, 46, 49] [2, 11, 13, 15, 21, 27, 30, 40, 45, 50] [1, 8, 12, 14, 22, 24, 28, 41, 44, 47] [2, 7, 14, 16, 18, 24, 33, 43, 50, 53] [3, 6, 15, 17, 19, 25, 34, 44, 49, 54] [0, 5, 12, 16, 18, 26, 31, 45, 48, 51] [0, 6, 11, 18, 20, 22, 29, 37, 47, 54] [2, 4, 9, 16, 20, 22, 28, 35, 49, 52] [4, 10, 15, 22, 24, 26, 30, 33, 41, 51]

Code ID 55-14-7 · download JSON · raw on GitHub