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[[51,12,3]] d =
n
51
k
12
d
3
kd²/n
2.118
w
7

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Distance

d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[6, 8, 10]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[8, 20, 32]
certificate exact, d = 3 · scipy/HiGHS MILP
X: no logical < 3 exists; Z: no logical < 3 exists

Construction & provenance

provenance submitted through the challenge
novelty novelty not audited
construction Hypergraph product of classical codes ham7 and sham6
model Tencent Hy3 (claimed, not verified)
builds on arXiv:0903.0566
date 2026-07-17
family hypergraph product (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 18 · Z-checks 21
H_X (18 checks, sparse supports)
[0, 12, 24, 36, 43] [1, 13, 25, 37, 42, 43] [2, 14, 26, 38, 44] [3, 15, 27, 39, 42, 44] [4, 16, 28, 40, 43, 44] [5, 17, 29, 41, 42, 43, 44] [6, 12, 30, 36, 46] [7, 13, 31, 37, 45, 46] [8, 14, 32, 38, 47] [9, 15, 33, 39, 45, 47] [10, 16, 34, 40, 46, 47] [11, 17, 35, 41, 45, 46, 47] [18, 24, 30, 36, 49] [19, 25, 31, 37, 48, 49] [20, 26, 32, 38, 50] [21, 27, 33, 39, 48, 50] [22, 28, 34, 40, 49, 50] [23, 29, 35, 41, 48, 49, 50]
H_Z (21 checks, sparse supports)
[1, 3, 5, 42] [0, 1, 4, 5, 43] [2, 3, 4, 5, 44] [7, 9, 11, 45] [6, 7, 10, 11, 46] [8, 9, 10, 11, 47] [13, 15, 17, 42, 45] [12, 13, 16, 17, 43, 46] [14, 15, 16, 17, 44, 47] [19, 21, 23, 48] [18, 19, 22, 23, 49] [20, 21, 22, 23, 50] [25, 27, 29, 42, 48] [24, 25, 28, 29, 43, 49] [26, 27, 28, 29, 44, 50] [31, 33, 35, 45, 48] [30, 31, 34, 35, 46, 49] [32, 33, 34, 35, 47, 50] [37, 39, 41, 42, 45, 48] [36, 37, 40, 41, 43, 46, 49] [38, 39, 40, 41, 44, 47, 50]