[[79,27,6]] · QEC Challenge

[[79,27,6]] d =

kd²/n

12.3038

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 · 2026-11-01

witness operator (support, 6 qubits)

[15, 23, 25, 40, 55, 58]

d_Z 6 · witness weight 6 (claimed upper_bound)

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

witness operator (support, 6 qubits)

[0, 6, 12, 30, 64, 73]

certificate exact, d = 6 · 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 < 6 exists; Z: no logical < 6 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 10 · H_Z 10

qubit degrees H_X 1–5 (mean 3.291) · H_Z 1–5 (mean 3.291)

trapping sets H_X (1,1)×4 (2,1)×3 (3,1)×5 (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): 5 (1,3): 41 (1,4): 22 (1,5): 7 (2,1): 3 (2,2): 25 (2,3): 92 (2,4): 308 (2,5): 329 (2,6): 233 (2,7): 73 (2,8): 11 (3,1): 5 (3,2): 86 (3,3): 486 (3,4): 1525 (3,5): 3635 (3,6): 5058 (3,7): 4898 (3,8): 3044 (3,9): 1325 (3,10): 346 (3,11): 93 (3,12): 15

trapping sets H_Z (1,1)×1 (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_Z

(1,1): 1 (1,2): 19 (1,3): 25 (1,4): 24 (1,5): 10 (2,1): 2 (2,2): 29 (2,3): 126 (2,4): 226 (2,5): 328 (2,6): 227 (2,7): 108 (2,8): 24 (3,1): 11 (3,2): 92 (3,3): 515 (3,4): 1585 (3,5): 3341 (3,6): 4545 (3,7): 4645 (3,8): 3396 (3,9): 1726 (3,10): 632 (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-PUNCTURE transform of codes/80-26-7.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

[[79,27,6]] — priced calculus E-PUNCTURE transform of 80-26-7.json

Direction & hypothesis

Weight-10 unrestricted CSS cell. Transform a connected CSS parent (80-26-7.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/80-26-7.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/80-26-7.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/80-26-7.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 26 (max weight 10) · Z-checks 26 (max weight 10)

H_X (26 checks, sparse supports)

[0, 14, 18, 24, 39, 41, 51, 61, 64, 71] [7, 8, 22, 26, 32, 40, 49, 59, 68, 71] [5, 10, 21, 27, 39, 45, 53, 60, 69, 71] [6, 10, 16, 31, 32, 43, 53, 56, 64, 72] [1, 15, 19, 25, 38, 40, 50, 60, 65, 72] [4, 11, 20, 26, 38, 44, 52, 61, 70, 72] [3, 9, 22, 25, 39, 44, 49, 57, 64, 73] [2, 8, 23, 24, 38, 45, 48, 56, 65, 74] [3, 13, 17, 27, 36, 42, 48, 62, 66, 74] [4, 11, 21, 25, 35, 43, 50, 56, 69, 74] [3, 12, 18, 30, 36, 44, 53, 63, 65, 75] [4, 10, 22, 28, 35, 45, 55, 57, 67, 75] [3, 15, 21, 26, 37, 46, 48, 61, 68, 75] [5, 15, 16, 31, 33, 42, 55, 63, 69, 75] [0, 15, 16, 30, 34, 40, 48, 57, 66, 76] [2, 14, 20, 27, 36, 47, 49, 60, 67, 76] [5, 11, 23, 29, 34, 44, 54, 56, 68, 76] [3, 12, 19, 29, 33, 43, 51, 58, 64, 77] [1, 14, 16, 28, 38, 46, 55, 61, 66, 77] [7, 13, 18, 29, 35, 40, 53, 61, 67, 77] [6, 8, 20, 30, 33, 47, 53, 59, 69, 77] [1, 13, 23, 24, 39, 44, 50, 63, 70, 77] [2, 13, 18, 28, 32, 42, 50, 59, 65, 78] [6, 12, 19, 28, 34, 41, 52, 60, 68, 78] [0, 12, 22, 25, 38, 45, 51, 62, 69, 78] [7, 9, 21, 31, 32, 46, 52, 58, 70, 78]

H_Z (26 checks, sparse supports)

[7, 14, 22, 27, 37, 46, 49, 62, 67, 71] [5, 11, 17, 24, 32, 42, 54, 56, 70, 71] [7, 10, 20, 30, 39, 47, 53, 57, 70, 71] [0, 10, 19, 27, 38, 45, 51, 60, 66, 72] [6, 15, 23, 26, 36, 47, 48, 63, 68, 72] [6, 11, 21, 31, 38, 46, 52, 56, 69, 72] [4, 10, 16, 25, 33, 43, 55, 57, 69, 72] [1, 9, 20, 26, 32, 46, 49, 59, 70, 72] [3, 9, 16, 24, 37, 46, 48, 63, 64, 73] [2, 10, 23, 25, 35, 45, 50, 56, 67, 73] [7, 9, 19, 26, 34, 40, 52, 58, 68, 73] [4, 9, 23, 29, 36, 44, 54, 58, 67, 74] [6, 8, 18, 27, 35, 41, 53, 59, 67, 74] [3, 11, 22, 24, 34, 44, 51, 57, 68, 74] [4, 13, 21, 24, 38, 45, 50, 61, 70, 74] [3, 14, 16, 26, 35, 43, 49, 61, 66, 75] [5, 15, 22, 30, 35, 40, 54, 57, 69, 75] [2, 11, 19, 30, 32, 43, 52, 59, 65, 76] [5, 13, 16, 30, 36, 42, 53, 63, 66, 76] [4, 14, 23, 31, 34, 41, 55, 56, 70, 76] [6, 14, 19, 29, 39, 41, 54, 60, 64, 77] [1, 12, 18, 24, 33, 41, 51, 63, 65, 77] [7, 13, 20, 28, 33, 42, 52, 59, 67, 77] [0, 13, 19, 25, 32, 40, 50, 62, 64, 78] [2, 12, 22, 31, 39, 45, 49, 63, 64, 78] [0, 9, 17, 28, 34, 41, 54, 57, 66, 78]

Code ID 79-27-6 · download JSON · raw on GitHub