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[[54,6,9]] d =
n
54
k
6
d
9
kd²/n
9.0
w
8

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Distance

d_X 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[0, 7, 8, 13, 25, 28, 46, 52, 53]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[2, 5, 12, 13, 18, 29, 33, 35, 42]
certificate exact, d = 9 · scipy/HiGHS MILP
X: no logical < 9 exists; Z: no logical < 9 exists

Construction & provenance

authors Lin, Hsiang-Ku and Pryadko, Leonid P.
provenance literature baseline
construction Two-block group algebra (2BGA) code over C_27: a={1, r, r3, r7}, b={1, r, r12, r19} (Lin & Pryadko, arXiv:2306.16400, Table 1).
model classical construction (no AI model)
reference arXiv:2306.16400
date 2023
notes Literature baseline (arXiv:2306.16400, Lin & Pryadko, 28 Jun 2023). Two-block group algebra code reconstructed from the published group presentation; distance d=9 taken from the paper's Table 1 and certified by an explicit non-stabilizer witness on each side (regenerated via the research kit's RIS search). Literature novelty confirmed by manual cross-check: arXiv:2306.16400 Table 1 explicitly reports the [[54,6,9]] parameters with W=6 (reconstructed checks have weight 8). The verifier's 'literature novelty UNVERIFIED' label is by-design (validate_candidate.py performs no literature lookup) and is intentionally not overridden.
family 2BGA coset (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 27 · Z-checks 27
H_X (27 checks, sparse supports)
[0, 20, 24, 26, 27, 35, 42, 53] [0, 1, 21, 25, 27, 28, 36, 43] [1, 2, 22, 26, 28, 29, 37, 44] [0, 2, 3, 23, 29, 30, 38, 45] [1, 3, 4, 24, 30, 31, 39, 46] [2, 4, 5, 25, 31, 32, 40, 47] [3, 5, 6, 26, 32, 33, 41, 48] [0, 4, 6, 7, 33, 34, 42, 49] [1, 5, 7, 8, 34, 35, 43, 50] [2, 6, 8, 9, 35, 36, 44, 51] [3, 7, 9, 10, 36, 37, 45, 52] [4, 8, 10, 11, 37, 38, 46, 53] [5, 9, 11, 12, 27, 38, 39, 47] [6, 10, 12, 13, 28, 39, 40, 48] [7, 11, 13, 14, 29, 40, 41, 49] [8, 12, 14, 15, 30, 41, 42, 50] [9, 13, 15, 16, 31, 42, 43, 51] [10, 14, 16, 17, 32, 43, 44, 52] [11, 15, 17, 18, 33, 44, 45, 53] [12, 16, 18, 19, 27, 34, 45, 46] [13, 17, 19, 20, 28, 35, 46, 47] [14, 18, 20, 21, 29, 36, 47, 48] [15, 19, 21, 22, 30, 37, 48, 49] [16, 20, 22, 23, 31, 38, 49, 50] [17, 21, 23, 24, 32, 39, 50, 51] [18, 22, 24, 25, 33, 40, 51, 52] [19, 23, 25, 26, 34, 41, 52, 53]
H_Z (27 checks, sparse supports)
[0, 1, 12, 19, 27, 28, 30, 34] [1, 2, 13, 20, 28, 29, 31, 35] [2, 3, 14, 21, 29, 30, 32, 36] [3, 4, 15, 22, 30, 31, 33, 37] [4, 5, 16, 23, 31, 32, 34, 38] [5, 6, 17, 24, 32, 33, 35, 39] [6, 7, 18, 25, 33, 34, 36, 40] [7, 8, 19, 26, 34, 35, 37, 41] [0, 8, 9, 20, 35, 36, 38, 42] [1, 9, 10, 21, 36, 37, 39, 43] [2, 10, 11, 22, 37, 38, 40, 44] [3, 11, 12, 23, 38, 39, 41, 45] [4, 12, 13, 24, 39, 40, 42, 46] [5, 13, 14, 25, 40, 41, 43, 47] [6, 14, 15, 26, 41, 42, 44, 48] [0, 7, 15, 16, 42, 43, 45, 49] [1, 8, 16, 17, 43, 44, 46, 50] [2, 9, 17, 18, 44, 45, 47, 51] [3, 10, 18, 19, 45, 46, 48, 52] [4, 11, 19, 20, 46, 47, 49, 53] [5, 12, 20, 21, 27, 47, 48, 50] [6, 13, 21, 22, 28, 48, 49, 51] [7, 14, 22, 23, 29, 49, 50, 52] [8, 15, 23, 24, 30, 50, 51, 53] [9, 16, 24, 25, 27, 31, 51, 52] [10, 17, 25, 26, 28, 32, 52, 53] [0, 11, 18, 26, 27, 29, 33, 53]
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