[[84,34,6]] · QEC Challenge

[[84,34,6]] d =

kd²/n

14.571

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 12, w_Z = 12 (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 exact)

witness operator (support, 6 qubits)

[0, 2, 13, 28, 36, 76]

d_Z 6 · witness weight 6 (claimed exact)

witness operator (support, 6 qubits)

[0, 18, 27, 37, 68, 72]

certificate exact, d = 6 · CryptoMiniSat 5.14 SAT

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 12 · H_Z 12

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×84 (2,2)×28 (3,4)×1316 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 84 (2,2): 28 (2,4): 196 (2,6): 1372 (3,4): 1316 (3,6): 9968 (3,8): 25452 (3,10): 2296

trapping sets H_Z (1,4)×84 (2,2)×28 (3,4)×1316 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 84 (2,2): 28 (2,4): 196 (2,6): 1372 (3,4): 1316 (3,6): 9968 (3,8): 25452 (3,10): 2296

Construction & provenance

authors Okada, Koki and Kasai, Kenta

provenance literature baseline

construction Cyclic all-one pair-partition CPM CSS construction (arXiv:2607.14091), small-p frontier instance qc_84_34_6 (internal id j4_l12_p7_small_frontier_s5): lift size p=7, (J,L)=(4,12), n=12*7=84, 4*7=28 checks per side, column weight 4, row weight 12. CPM exponent arrays (rows = the four CPM block-rows per side, columns = the 12 block-columns): E_X = [[0, 4, 0, 4, 5, 6, 2, 4, 5, 0, 4, 6], [5, 1, 6, 3, 5, 1, 3, 1, 6, 3, 0, 3], [3, 0, 5, 0, 2, 6, 3, 1, 6, 6, 3, 6], [4, 3, 2, 4, 1, 4, 2, 1, 6, 1, 2, 1]], E_Z = [[4, 0, 1, 6, 5, 5, 6, 0, 6, 2, 4, 5], [1, 6, 0, 4, 2, 0, 2, 5, 0, 4, 5, 6], [1, 3, 5, 0, 6, 1, 5, 5, 6, 6, 6, 4], [4, 4, 5, 6, 5, 2, 2, 2, 2, 3, 6, 6]]. Lift convention per the authors' expand_and_verify_cpm_matrices.py (github.com/kasaikenta/pair-partition-cpm-css-codes @ 9c2a6f2): check a of block-row b on side S has support {12*((a - E_S[b][t]) mod 7) + t : t=0..11}. Source: https://kasai.ict.eng.isct.ac.jp/pair_partition_cpm_css_codes_data/certificate_bundles_20260805/qc_84_34_6/certificate_bundle.tar.gz, artifacts/fixed_input/candidate.cpm (sha256 abb5eeef2c2d99f4be5395fafad4aa3d4c8fb309980a28fd4fda6630d86a781d), expanded with the authors' own script.

model classical construction (no AI model)

reference arXiv:2607.14091

date 2026

notes Literature baseline, seeded 2026-08-14. Published in the certificate-bundle set of the Okada-Kasai pair-partition CPM catalogue (arXiv:2607.14091; certificate_bundles_20260805, directory qc_84_34_6, status exact_distance_machine_verified in master_manifest.json). The authors' bundle gives complete lower-bound exclusion through weight 5 on both sides plus explicit weight-6 logicals, certifying d=6 exactly; the witnesses here are the bundle's published upper_logical_vectors.json entries, re-verified in this repo (kernel membership, row-space non-membership). Marked exact on the strength of that public machine-verified bundle (complete lower-bound exclusion through weight 5 on both sides, SHA-256-pinned artifacts, independent-verifier record included). Per board policy the entry still displays as an upper bound until this repo's own certification. Reproduction was validated end to end: the bundle's fixed-input .cpm expanded with the authors' own script gives a CSS-orthogonal (4,12)-regular pair with GF(2) ranks 25/25 (k=34).

family pair-partition CPM (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight > 8 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 28 (max weight 12) · Z-checks 28 (max weight 12)

H_X (28 checks, sparse supports)

[0, 2, 9, 17, 23, 28, 32, 37, 39, 43, 46, 66] [12, 14, 21, 29, 35, 40, 44, 49, 51, 55, 58, 78] [6, 24, 26, 33, 41, 47, 52, 56, 61, 63, 67, 70] [18, 36, 38, 45, 53, 59, 64, 68, 73, 75, 79, 82] [1, 3, 7, 10, 30, 48, 50, 57, 65, 71, 76, 80] [4, 8, 13, 15, 19, 22, 42, 60, 62, 69, 77, 83] [5, 11, 16, 20, 25, 27, 31, 34, 54, 72, 74, 81] [10, 14, 20, 24, 28, 51, 54, 57, 59, 73, 77, 79] [1, 5, 7, 22, 26, 32, 36, 40, 63, 66, 69, 71] [13, 17, 19, 34, 38, 44, 48, 52, 75, 78, 81, 83] [3, 6, 9, 11, 25, 29, 31, 46, 50, 56, 60, 64] [15, 18, 21, 23, 37, 41, 43, 58, 62, 68, 72, 76] [0, 4, 27, 30, 33, 35, 49, 53, 55, 70, 74, 80] [2, 8, 12, 16, 39, 42, 45, 47, 61, 65, 67, 82] [1, 3, 17, 20, 21, 23, 26, 48, 54, 58, 64, 79] [7, 13, 15, 29, 32, 33, 35, 38, 60, 66, 70, 76] [4, 19, 25, 27, 41, 44, 45, 47, 50, 72, 78, 82] [0, 6, 10, 16, 31, 37, 39, 53, 56, 57, 59, 62] [12, 18, 22, 28, 43, 49, 51, 65, 68, 69, 71, 74] [2, 24, 30, 34, 40, 55, 61, 63, 77, 80, 81, 83] [5, 8, 9, 11, 14, 36, 42, 46, 52, 67, 73, 75] [20, 36, 39, 41, 49, 62, 66, 70, 76, 79, 81, 83] [4, 7, 9, 11, 32, 48, 51, 53, 61, 74, 78, 82] [2, 6, 10, 16, 19, 21, 23, 44, 60, 63, 65, 73] [1, 14, 18, 22, 28, 31, 33, 35, 56, 72, 75, 77] [0, 3, 5, 13, 26, 30, 34, 40, 43, 45, 47, 68] [12, 15, 17, 25, 38, 42, 46, 52, 55, 57, 59, 80] [8, 24, 27, 29, 37, 50, 54, 58, 64, 67, 69, 71]

H_Z (28 checks, sparse supports)

[1, 7, 15, 18, 20, 28, 29, 35, 36, 46, 69, 74] [2, 13, 19, 27, 30, 32, 40, 41, 47, 48, 58, 81] [9, 14, 25, 31, 39, 42, 44, 52, 53, 59, 60, 70] [21, 26, 37, 43, 51, 54, 56, 64, 65, 71, 72, 82] [0, 10, 33, 38, 49, 55, 63, 66, 68, 76, 77, 83] [4, 5, 11, 12, 22, 45, 50, 61, 67, 75, 78, 80] [3, 6, 8, 16, 17, 23, 24, 34, 57, 62, 73, 79] [2, 5, 8, 13, 23, 31, 34, 39, 45, 64, 66, 72] [0, 14, 17, 20, 25, 35, 43, 46, 51, 57, 76, 78] [4, 6, 12, 26, 29, 32, 37, 47, 55, 58, 63, 69] [16, 18, 24, 38, 41, 44, 49, 59, 67, 70, 75, 81] [3, 9, 28, 30, 36, 50, 53, 56, 61, 71, 79, 82] [7, 10, 15, 21, 40, 42, 48, 62, 65, 68, 73, 83] [1, 11, 19, 22, 27, 33, 52, 54, 60, 74, 77, 80] [3, 16, 20, 21, 22, 26, 30, 31, 47, 49, 72, 77] [0, 5, 15, 28, 32, 33, 34, 38, 42, 43, 59, 61] [12, 17, 27, 40, 44, 45, 46, 50, 54, 55, 71, 73] [1, 24, 29, 39, 52, 56, 57, 58, 62, 66, 67, 83] [11, 13, 36, 41, 51, 64, 68, 69, 70, 74, 78, 79] [2, 6, 7, 23, 25, 48, 53, 63, 76, 80, 81, 82] [4, 8, 9, 10, 14, 18, 19, 35, 37, 60, 65, 75] [15, 22, 23, 26, 28, 36, 37, 57, 65, 66, 67, 68] [27, 34, 35, 38, 40, 48, 49, 69, 77, 78, 79, 80] [5, 6, 7, 8, 39, 46, 47, 50, 52, 60, 61, 81] [9, 17, 18, 19, 20, 51, 58, 59, 62, 64, 72, 73] [0, 1, 21, 29, 30, 31, 32, 63, 70, 71, 74, 76] [2, 4, 12, 13, 33, 41, 42, 43, 44, 75, 82, 83] [3, 10, 11, 14, 16, 24, 25, 45, 53, 54, 55, 56]

Code ID 84-34-6 · download JSON · raw on GitHub