[[55,13,7]] · QEC Challenge

[[55,13,7]] d =

kd²/n

11.5818

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · w_X = 8, w_Z = 8 (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)

[9, 10, 14, 38, 39, 47, 49]

d_Z 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)

[10, 11, 18, 36, 38, 41, 42]

certificate exact, d = 7 · tropidist b03ec2d4077f (arXiv:2611.08247) exhaustive search · level solver: a solver verdict, no artifact a third party can check without re-running the encoding

X: no logical < 7 exists; Z: no logical < 7 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 8 · H_Z 8

qubit degrees H_X 1–5 (mean 3.055) · H_Z 1–5 (mean 3.055)

trapping sets H_X (1,1)×3 (2,1)×2 (3,1)×11 (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): 15 (1,3): 21 (1,4): 8 (1,5): 8 (2,1): 2 (2,2): 27 (2,3): 95 (2,4): 125 (2,5): 137 (2,6): 111 (2,7): 33 (2,8): 17 (3,1): 11 (3,2): 84 (3,3): 351 (3,4): 933 (3,5): 1424 (3,6): 1816 (3,7): 1479 (3,8): 915 (3,9): 487 (3,10): 179 (3,11): 44 (3,12): 4 (3,13): 2

trapping sets H_Z (1,1)×4 (2,1)×2 (3,1)×13 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 15 (1,3): 18 (1,4): 10 (1,5): 8 (2,1): 2 (2,2): 29 (2,3): 87 (2,4): 114 (2,5): 148 (2,6): 100 (2,7): 50 (2,8): 15 (3,1): 13 (3,2): 76 (3,3): 356 (3,4): 872 (3,5): 1368 (3,6): 1728 (3,7): 1524 (3,8): 1030 (3,9): 516 (3,10): 193 (3,11): 42 (3,12): 13

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PUNCTURE transform of codes/56-12-8.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

[[55,13,7]] — priced calculus E-PUNCTURE transform of 56-12-8.json

Direction & hypothesis

Weight-8 unrestricted CSS cell. Transform a connected CSS parent (56-12-8.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/56-12-8.json. Rule used for this survivor: E-PUNCTURE. 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/56-12-8.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/56-12-8.json by applying the E-PUNCTURE move of the priced calculus as recorded above; re-run the witness search at the stated budget.

Parity checks

X-checks 21 (max weight 8) · Z-checks 21 (max weight 8)

H_X (21 checks, sparse supports)

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

H_Z (21 checks, sparse supports)

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

Code ID 55-13-7 · download JSON · raw on GitHub