[[70,8,7]] · QEC Challenge

[[70,8,7]] d ≤

kd²/n

5.6

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (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, 7 qubits)

[8, 12, 29, 31, 55, 58, 61]

d_Z 7 · witness weight 7 (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, 7 qubits)

[15, 24, 31, 37, 43, 45, 54]

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 5–6 (mean 5.935) · H_Z 5–6 (mean 5.935)

qubit degrees H_X 1–3 (mean 2.629) · H_Z 1–3 (mean 2.629)

trapping sets H_X (1,1)×3 (2,1)×1 (3,1)×8 (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): 20 (1,3): 47 (2,1): 1 (2,2): 33 (2,3): 158 (2,4): 263 (3,1): 8 (3,2): 62 (3,3): 460 (3,4): 1464 (3,5): 1917 (3,6): 240 (3,7): 259

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

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

(1,1): 3 (1,2): 20 (1,3): 47 (2,1): 1 (2,2): 33 (2,3): 158 (2,4): 263 (3,1): 8 (3,2): 62 (3,3): 460 (3,4): 1464 (3,5): 1917 (3,6): 240 (3,7): 259

Construction & provenance

authors @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Priced construction calculus E-PEEL of codes/71-8-7.json ([[71,8,7]], max check weight 6; itself a priced-calculus E-PEEL of codes/72-8-8.json, a generalized-bicycle entry isomorphic to a Lin and Pryadko 2BGA code over SmallGroup(36,8)): delete qubit 34 (0-based index in the parent file) together with its unique Z check b9, 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:2306.16400

date 2026-11-04

notes Single-step priced-calculus E-PEEL child of board entry 71-8-7. 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

[[70,8,7]] priced calculus E-PEEL of codes/71-8-7.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/71-8-7.json ([[71,8,7]], max check weight 6; itself a priced-calculus E-PEEL of codes/72-8-8.json, a generalized-bicycle entry isomorphic to a Lin and Pryadko 2BGA code over SmallGroup(36,8)). Peeling qubit 34 with the one Z check that touches it keeps k = 8 and the witnessed distance bound d <= 7 while dropping one qubit, so it sits one qubit to the left of its parent on the frontier. 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 [[70,8,7]] at max check weight 6. 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) | 7 | | 1,000,000 (seed 101) | 7 | | 1,000,000 (seed 102) | 7 | | 10,000,000 (seed 103) | 7 |

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 <= 7 (upper_bound), Z: d <= 7 (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/71-8-7.json, delete qubit 34 (0-based index in that file) together with its unique Z check b9, where the Z rows are numbered b1, b2, ... 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': 34, 'sector': 'Z', 'row_id': 'b9'}). Then run research/audits/leader_audit.py ladder on the result with --ladder 1000000:101,102 to reproduce the confirmation rungs; the witnesses in the entry are the ones those runs validated.

Parity checks

X-checks 31 (max weight 6) · Z-checks 31 (max weight 6)

H_X (31 checks, sparse supports)

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

H_Z (31 checks, sparse supports)

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

Code ID 70-8-7 · download JSON · raw on GitHub