[[54,18,5]] · QEC Challenge

[[54,18,5]] d =

kd²/n

8.33333

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 5, 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 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, 11, 29, 41, 44]

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 · 2026-11-01

witness operator (support, 5 qubits)

[5, 15, 18, 36, 49]

certificate exact, d = 5 · CryptoMiniSat 5.14 SAT · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI

X: no logical < 5 exists; Z: no logical < 5 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 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 9.778) · H_Z 9–10 (mean 9.833)

qubit degrees H_X 2–6 (mean 3.259) · H_Z 1–6 (mean 3.278)

trapping sets H_X (1,2)×16 (2,1)×2 (3,1)×17 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 16 (1,3): 18 (1,4): 13 (1,5): 4 (1,6): 3 (2,1): 2 (2,2): 32 (2,3): 97 (2,4): 166 (2,5): 178 (2,6): 107 (2,7): 54 (2,8): 14 (2,9): 3 (2,10): 1 (3,1): 17 (3,2): 125 (3,3): 498 (3,4): 1306 (3,5): 2151 (3,6): 2305 (3,7): 1893 (3,8): 1075 (3,9): 499 (3,10): 160 (3,11): 43 (3,12): 7

trapping sets H_Z (1,1)×1 (2,1)×4 (3,1)×14 (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): 15 (1,3): 16 (1,4): 15 (1,5): 4 (1,6): 3 (2,1): 4 (2,2): 27 (2,3): 83 (2,4): 187 (2,5): 160 (2,6): 120 (2,7): 52 (2,8): 19 (2,9): 4 (2,10): 1 (3,1): 14 (3,2): 104 (3,3): 505 (3,4): 1282 (3,5): 2028 (3,6): 2386 (3,7): 1907 (3,8): 1153 (3,9): 533 (3,10): 200 (3,11): 45 (3,12): 9 (3,13): 1

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PEEL transform of codes/56-18-6.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

[[54,18,5]] — priced calculus E-PEEL transform of 56-18-6.json

Direction & hypothesis

Weight-10 unrestricted CSS cell. Transform a connected CSS parent (56-18-6.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/56-18-6.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/56-18-6.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/56-18-6.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 18 (max weight 10) · Z-checks 18 (max weight 10)

H_X (18 checks, sparse supports)

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

H_Z (18 checks, sparse supports)

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

Code ID 54-18-5 · download JSON · raw on GitHub