[[70,12,5]] · QEC Challenge

[[70,12,5]] d ≤

kd²/n

4.28571

X/Z

1.2

Share this result

Distance

X/Z asymmetry 1.2 · d_X ≤ 6, d_Z ≤ 5 · w_X = 6, w_Z = 6 (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 @vprusso · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-03

witness operator (support, 6 qubits)

[40, 42, 59, 60, 61, 62]

d_Z 5 · witness weight 5 (claimed upper_bound)

witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-03

witness operator (support, 5 qubits)

[14, 16, 29, 30, 31]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 4–6 (mean 5.833)

qubit degrees H_X 1–3 (mean 2.4) · H_Z 1–3 (mean 2.5)

trapping sets H_X (1,1)×8 (2,1)×7 (3,1)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 8 (1,2): 26 (1,3): 36 (2,1): 7 (2,2): 71 (2,3): 177 (2,4): 165 (3,1): 24 (3,2): 187 (3,3): 795 (3,4): 1353 (3,5): 1120 (3,6): 230 (3,7): 125

trapping sets H_Z (1,1)×7 (2,1)×5 (3,1)×15 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 7 (1,2): 21 (1,3): 42 (2,1): 5 (2,2): 49 (2,3): 162 (2,4): 210 (3,1): 15 (3,2): 120 (3,3): 576 (3,4): 1381 (3,5): 1437 (3,6): 240 (3,7): 175

Construction & provenance

authors @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Priced construction calculus E-PEEL of codes/71-12-5.json ([[71,12,5]], max check weight 6; itself Mathys Rennela's priced-calculus E-PEEL of codes/72-12-6.json, the [[72,12,6]] bivariate bicycle code of arXiv:2308.07915): delete qubit 29 (0-based index in the parent file) together with its unique X check a12, and relabel the surviving qubits in increasing order of their parent indices. Consumer: github.com/unitaryfoundation/code-calculus @ 1b9a942, machine/priced_calculus.py.

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:2308.07915

date 2026-11-04

notes Single-step priced-calculus E-PEEL child of board entry 71-12-5. verify/validate_candidate.py found no board entry sharing this code's fingerprint or Weisfeiler-Leman signature, local-Hadamard images included.

family other (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[70,12,5]] priced calculus E-PEEL of codes/71-12-5.json

Direction & hypothesis

Unrestricted weight-6 CSS cell. The priced construction calculus was applied one move deep to every CSS entry on the board, and this code is the E-PEEL of codes/71-12-5.json ([[71,12,5]], max check weight 6; itself Mathys Rennela's priced-calculus E-PEEL of codes/72-12-6.json, the [[72,12,6]] bivariate bicycle code of arXiv:2308.07915). Peeling qubit 29 with the one X check that touches it keeps k = 12 and keeps the witnessed distance bound at 5 while dropping one qubit. Max check weight stays 6.

What was searched

Every C-DELETE-CHECK, E-PEEL, and E-PUNCTURE move on each of the 2,783 CSS board entries as of 2026-11-02, 1,684,787 moves in all, with the consumer at github.com/unitaryfoundation/code-calculus @ 1b9a942. Children with a bare qubit, a disconnected Tanner graph, a Weisfeiler-Leman signature equal to a board entry or to an earlier child, or parameters already dominated inside the parent's cell were dropped before any search, leaving 334,191 distinct children. Children with n <= 60 and an inherited distance bound of at least 2, and children with 61 <= n <= 100 and a bound of at least 5, were screened with research/kit/surrogate.lightest_logical at 2,000 RIS trials per side; the survivors the kit's novelty pre-check called board-advancing were staged with research/kit/submit.py and gated with verify/validate_candidate.py, one representative per distinct (n, k, d, max weight, locality) point. This code is that representative for [[70,12,5]] at max check weight 6. The thirteen points audited first went to the board on 2026-11-04 (PRs #4588 to #4600); this one is from the remainder, re-checked against the board of 2026-11-04 with the gate's structural, deduplication, and novelty stages before the ladder below was run. The screen distance is a witness-backed upper bound; the rungs below are the independent confirmation runs on the staged file.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 2,000 (staging screen, seeds 1 and 2) | 5 | | 1,000,000 (seed 101) | 5 | | 1,000,000 (seed 102) | 5 | | 10,000,000 (seed 103) | 5 |

4 fresh-seed rung(s) agree at the best bound, the deepest of them at 10,000,000 trials per side. The submitted witness reproduces it: X: d <= 6 (upper_bound), Z: d <= 5 (upper_bound).

Dead ends

None recorded for this candidate.

Tools

Model: Claude Fable 5.1 (Claude Code). Repo tooling: research/kit/coordination.py, research/kit/submit.py, research/kit/surrogate.py, research/audits/leader_audit.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py.

Reproduction

Load codes/71-12-5.json, delete qubit 29 (0-based index in that file) together with its unique X check a12, where the X rows are numbered a1, a2, ... in file order after dropping rows dependent on earlier ones, as CodeState.from_challenge in machine/priced_calculus.py labels them, and relabel the surviving qubits in increasing order of their parent indices; in github.com/unitaryfoundation/code-calculus @ 1b9a942 this is apply_operation(state, {'op': 'E-PEEL', 'qubit': 29, 'sector': 'X', 'row_id': 'a12'}). Then run research/audits/leader_audit.py ladder on the result with --ladder 1000000:101,102 10000000:103 to reproduce the confirmation rungs; the witnesses in the entry are the ones those runs validated.

Parity checks

X-checks 28 (max weight 6) · Z-checks 30 (max weight 6)

H_X (28 checks, sparse supports)

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

H_Z (30 checks, sparse supports)

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

Code ID 70-12-5 · download JSON · raw on GitHub