← back to the board
[[83,14,6]] d ≤
n
83
k
14
d
6
kd²/n
6.07229
w
7
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 7, w_Z = 7 (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)
[44, 46, 65, 68, 71, 74]
d_Z 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)
[26, 29, 37, 39, 40, 41]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 7 · H_Z 6–7 (mean 6.914)
qubit degrees H_X 1–4 (mean 2.867) · H_Z 1–4 (mean 2.916)
trapping sets H_X (1,1)×2 (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): 2 (1,2): 24 (1,3): 40 (1,4): 17 (2,1): 2 (2,2): 30 (2,3): 172 (2,4): 268 (2,5): 182 (2,6): 33 (3,1): 11 (3,2): 101 (3,3): 528 (3,4): 1586 (3,5): 2520 (3,6): 2234 (3,7): 1065 (3,8): 276 (3,9): 71 (3,10): 4
trapping sets H_Z (1,1)×3 (2,1)×2 (3,1)×7 (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): 19 (1,3): 43 (1,4): 18 (2,1): 2 (2,2): 23 (2,3): 143 (2,4): 287 (2,5): 197 (2,6): 37 (3,1): 7 (3,2): 90 (3,3): 412 (3,4): 1380 (3,5): 2550 (3,6): 2460 (3,7): 1177 (3,8): 315 (3,9): 80 (3,10): 4

Circuit tier

syndrome-extraction memory circuits committed under circuits/83-14-6/ · canonical noise recipe, 6 rounds, stim 1.16.0
d_circ ≤ 6 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 6 · fault-set witness of 6 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 6)
[354, 359, 427, 454, 497, 501]
d_circ^Z 6 · fault-set witness of 6 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 6)
[33216, 33220, 33222, 33226, 33233, 33244]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Priced construction calculus E-PEEL of codes/84-14-6.json ([[84,14,6]], max check weight 7; a two-block group-algebra code over SmallGroup(42,3) from arXiv:2610.13880): delete qubit 64 (0-based index in the parent file) together with its unique X check a23, 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 Claude Fable 5.1 (claimed, not verified)
builds on arXiv:2610.13880
date 2026-11-04
notes Single-step priced-calculus E-PEEL child of board entry 84-14-6. verify/validate_candidate.py found no board entry sharing this code's fingerprint or Weisfeiler-Leman signature, local-Hadamard images included.
family 2bga-nonabelian (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

[[83,14,6]] priced calculus E-PEEL of codes/84-14-6.json

Direction & hypothesis

Unrestricted weight-8 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/84-14-6.json ([[84,14,6]], max check weight 7; a two-block group-algebra code over SmallGroup(42,3) from arXiv:2610.13880). Peeling qubit 64 with the one X check that touches it keeps k = 14 and keeps the witnessed distance bound at 6 while dropping one qubit. Max check weight stays 7.

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 [[83,14,6]] at max check weight 7. 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) | 6 | | 1,000,000 (seed 101) | 6 | | 1,000,000 (seed 102) | 6 | | 10,000,000 (seed 103) | 6 |

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 <= 6 (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/84-14-6.json, delete qubit 64 (0-based index in that file) together with its unique X check a23, 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': 64, 'sector': 'X', 'row_id': 'a23'}). 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 34 (max weight 7) · Z-checks 35 (max weight 7)
H_X (34 checks, sparse supports)
[0, 4, 15, 30, 42, 66, 81] [1, 2, 12, 33, 43, 69, 82] [0, 2, 9, 21, 44, 50, 72] [3, 10, 22, 36, 45, 71, 73] [1, 4, 6, 18, 46, 53, 75] [5, 7, 19, 38, 47, 74, 76] [2, 6, 15, 27, 48, 56, 77] [3, 7, 16, 28, 42, 49, 78] [8, 17, 29, 40, 50, 60, 77] [4, 9, 12, 24, 51, 59, 79] [5, 10, 13, 25, 43, 52, 80] [11, 14, 26, 41, 53, 63, 79] [6, 12, 21, 33, 45, 54, 62] [7, 13, 22, 34, 44, 55, 81] [8, 14, 23, 35, 45, 56, 65] [9, 15, 18, 30, 47, 57, 64] [10, 16, 19, 31, 46, 58, 82] [11, 17, 20, 32, 47, 59, 68] [1, 12, 18, 27, 49, 60, 67] [13, 19, 28, 38, 48, 50, 61] [14, 20, 29, 39, 49, 62, 71] [0, 15, 21, 24, 52, 63, 70] [17, 23, 26, 37, 52, 64, 74] [4, 18, 24, 33, 55, 65, 73] [5, 19, 25, 34, 54, 56, 66] [20, 26, 35, 41, 42, 55, 67] [2, 21, 27, 30, 58, 68, 76] [3, 22, 28, 31, 57, 59, 69] [23, 29, 32, 40, 43, 58, 70] [1, 9, 24, 30, 61, 71, 78] [10, 25, 31, 38, 60, 62, 72] [11, 26, 32, 39, 44, 61, 73] [7, 28, 34, 36, 63, 64, 75] [17, 32, 37, 41, 48, 66, 78]
H_Z (35 checks, sparse supports)
[0, 7, 26, 42, 44, 63, 74] [1, 10, 29, 43, 46, 60, 71] [2, 13, 32, 43, 44, 48, 68] [3, 12, 14, 45, 49, 69, 79] [4, 16, 35, 42, 46, 51, 65] [5, 15, 17, 47, 52, 66, 77] [6, 19, 37, 46, 48, 54, 74] [7, 18, 20, 47, 49, 55, 75] [2, 8, 19, 50, 56, 76, 82] [9, 22, 39, 44, 51, 57, 71] [10, 21, 23, 45, 52, 58, 72] [4, 11, 22, 53, 59, 73, 81] [12, 25, 40, 43, 51, 54, 60] [13, 24, 26, 52, 55, 61, 79] [6, 14, 25, 53, 56, 62, 80] [15, 28, 41, 42, 48, 57, 63] [16, 27, 29, 49, 58, 77] [9, 17, 28, 50, 59, 64, 78] [8, 18, 31, 46, 57, 60, 65] [19, 30, 32, 47, 58, 61, 66] [12, 20, 31, 59, 62, 67, 82] [11, 21, 34, 44, 54, 63, 68] [22, 33, 35, 45, 55, 69] [15, 23, 34, 56, 64, 70, 81] [14, 24, 36, 51, 63, 65, 71] [0, 25, 37, 52, 66, 72] [18, 26, 36, 53, 64, 67, 73] [17, 27, 38, 48, 60, 68, 74] [1, 28, 39, 49, 61, 69, 75] [21, 29, 38, 50, 62, 70, 76] [3, 20, 30, 42, 57, 68, 71] [2, 31, 40, 58, 69, 72, 77] [3, 24, 32, 59, 70, 73, 78] [4, 34, 41, 55, 66, 75, 79] [7, 30, 37, 64, 76, 78, 81]
Code ID 83-14-6 · download JSON · raw on GitHub