[[70,6,9]] · QEC Challenge

[[70,6,9]] d =

kd²/n

6.94286

X/Z

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Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · 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 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[7, 9, 23, 34, 37, 38, 43, 50, 63]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[13, 18, 31, 37, 51, 58, 61, 65, 68]

certificate exact, d = 9 · scipy/HiGHS cutoff IP · level proof_log: the refutation was re-checked by drat-trim and is replayed by CI

X: no logical < 9 exists; Z: no logical < 9 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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×35 (2,3)×35 (3,2)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 35 (1,4): 35 (2,3): 35 (2,4): 140 (2,5): 350 (2,6): 140 (3,2): 35 (3,3): 35 (3,4): 350 (3,5): 1225 (3,6): 2870 (3,7): 2940 (3,8): 980 (3,9): 280 (3,10): 35

trapping sets H_Z (1,3)×35 (2,3)×35 (3,2)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 35 (1,4): 35 (2,3): 35 (2,4): 140 (2,5): 350 (2,6): 140 (3,2): 35 (3,3): 35 (3,4): 350 (3,5): 1225 (3,6): 2870 (3,7): 2940 (3,8): 980 (3,9): 280 (3,10): 35

Construction & provenance

authors Tripier, Felix and Chung, Woo Chang and Young, Jacob and Alam, Safwan and Bjork, Bryce and Brodutch, Aharon and Buessen, Finn Lasse and Coble, Nolan J. and Dellaert, Thomas and Maslov, Dmitri and Roetteler, Martin and Tham, Edwin and Webster, Mark and Ye, Min and Gamble, John and Maksymov, Andrii and Marceaux, J. P. and Delfosse, Nicolas

provenance literature baseline

construction Bivariate bicycle code with l = 7, m = 5, x = S_7 (x) I_5, y = I_7 (x) S_5 (S the cyclic shift matrix), A = y² + x² + x³ + x⁴, B = y + x + x³, H_X = [A | B], H_Z = [B^T | A^T]; code Q70 of Table XXX of arXiv:2604.19481 (Tripier et al., the Walking Cat architecture). Rebuilt here from those polynomials; n, k, and the check weight 7 reproduce the paper's values.

model Claude Fable 5.1 (Claude Code) (claimed, not verified)

reference arXiv:2604.19481

date 2026-04-21

notes Literature baseline: a reproduction of code Q70 of arXiv:2604.19481 (Table XXX), not a new construction and not a new parameter set. Same parameters as 70-6-9, which reproduces a row of Lin and Pryadko's 2BGA database (July 2023), but a different code: each side of both codes has exactly 35 minimum-weight stabilizers, and nauty finds no isomorphism between the typed Tanner graphs on those complete sets, with or without an X/Z swap, so a signature match against 70-6-9 is two inequivalent codes with the same parameters. The paper reports d = 9; this entry carries witness-backed upper bounds from the board's own search until certified. Seeded after Nicolas Delfosse (IonQ) reported the code in review of the QEC Challenge paper.

family bivariate bicycle (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 35 (max weight 7) · Z-checks 35 (max weight 7)

H_X (35 checks, sparse supports)

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

H_Z (35 checks, sparse supports)

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

Code ID 70-6-9-b · download JSON · raw on GitHub