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[[252,130,6]] d ≤
n
252
k
130
d
6
kd²/n
18.571
w
12
X/Z
1

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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 upper_bound)
witness operator (support, 6 qubits)
[23, 125, 191, 194, 214, 235]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[75, 91, 96, 112, 116, 154]
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 12 · H_Z 12
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×252 (2,2)×210 (3,3)×5614 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 252 (2,2): 210 (2,4): 3738 (3,3): 5614 (3,5): 74214 (3,7): 9996
trapping sets H_Z (1,3)×252 (2,2)×210 (3,3)×5614 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 252 (2,2): 210 (2,4): 3738 (3,3): 5614 (3,5): 74214 (3,7): 9996

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Cornucopia code q=7 (arXiv:2610.19073v2) with column shifts sA=[0,4,6,1,0,0], sB=[0,2,4,3,4,4] found by a 200-tuple random shift search.
model Beam 1.0 (claimed, not verified)
date 2026-10-21
notes Distinct code filed as codes/252-130-6-b.json: WL-signature match to 252-130-6.json but a different parity-check fingerprint (different column-shift tuple of the same Cornucopia construction), so both are separate entries on the Pareto frontier.
family other (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

[[252,130,6]] — Cornucopia q=7 with searched column shifts

Direction & hypothesis

Target cell: weight-9plus × unrestricted on the CSS board (weight-12 checks, no layout), family other. The Cornucopia construction (Lu, Li, Ma et al., "A modular quantum computer based on Cornucopia codes", arXiv:2610.19073v2) fixes only the row maps; its twelve column shifts are free parameters. The hypothesis: at this admissible q (n = 36q ≤ 700) an unexplored shift tuple would produce a code that is not dominated by the Cornucopia instance of the same (n, k) already on the board.

What was searched

200-tuple random column-shift tuples at q=7 (sA, sB ∈ Z_7^6 each), every code screened with the gf2_fast randomized logical search at 500 trials, seeds from the sweep's own RNG stream. The best tuple, sA=[0, 4, 6, 1, 0, 0], sB=[0, 2, 4, 3, 4, 4], was then confirmed at 10,000 accelerator trials plus a 1,000-trial-per-side RIS pass.

Evidence trail

Ladder for the submitted code: 500-trial screen → d<=6; 10,000-trial accelerator pass → d<=6 (witnesses embedded in the entry); the gate's own refutation on a fresh seed found nothing lighter. The claim is a witness-backed upper bound, not an exact distance.

Dead ends

The same sweep at q=13 produced nothing above d ≤ 6 while the board already holds q=13 at d=8, so q=13 was dropped. Within this sweep, tuples screened at 500 trials routinely collapsed to d ≤ 4 at smaller q; none of the confirmed readings moved below the submitted bound.

Tools

Model: Beam 1.0. Repo tooling: research/kit/submit.make_submission, research/kit/surrogate (NumPy + gf2_fast backend), gate research/kit/coordination.py gate. Build matrices per arXiv:2610.19073v2 Methods, Eq. (1).

Reproduction

q=7, row maps ρ_A(0)=r+1, ρ_A(1)=2r+2, ρ_B(2)=r+1, ρ_B(3)=2r+2 (identity otherwise), column shifts sA=[0, 4, 6, 1, 0, 0], sB=[0, 2, 4, 3, 4, 4] on Z_7. Index data qubit (b,r,c) as (3b+r)q+c; HX block (i,j) = [A_(j-i) | B_(j-i)], HZ block (i,j) = [B_(i-j)^-1 | A_(i-j)^-1] for i∈{0,1,2}, j∈{0..5}.

Parity checks

X-checks 63 (max weight 12) · Z-checks 63 (max weight 12)
H_X (63 checks, sparse supports)
[7, 39, 48, 64, 84, 105, 126, 149, 179, 206, 214, 235] [8, 40, 42, 65, 85, 106, 127, 150, 180, 207, 215, 236] [9, 41, 43, 66, 86, 107, 128, 151, 181, 208, 216, 237] [10, 35, 44, 67, 87, 108, 129, 152, 175, 209, 210, 231] [11, 36, 45, 68, 88, 109, 130, 153, 176, 203, 211, 232] [12, 37, 46, 69, 89, 110, 131, 147, 177, 204, 212, 233] [13, 38, 47, 63, 90, 111, 132, 148, 178, 205, 213, 234] [14, 32, 55, 71, 91, 112, 133, 156, 186, 199, 221, 242] [15, 33, 49, 72, 92, 113, 134, 157, 187, 200, 222, 243] [16, 34, 50, 73, 93, 114, 135, 158, 188, 201, 223, 244] [17, 28, 51, 74, 94, 115, 136, 159, 182, 202, 217, 238] [18, 29, 52, 75, 95, 116, 137, 160, 183, 196, 218, 239] [19, 30, 53, 76, 96, 117, 138, 154, 184, 197, 219, 240] [20, 31, 54, 70, 97, 118, 139, 155, 185, 198, 220, 241] [0, 25, 62, 78, 98, 119, 140, 163, 172, 192, 228, 249] [1, 26, 56, 79, 99, 120, 141, 164, 173, 193, 229, 250] [2, 27, 57, 80, 100, 121, 142, 165, 174, 194, 230, 251] [3, 21, 58, 81, 101, 122, 143, 166, 168, 195, 224, 245] [4, 22, 59, 82, 102, 123, 144, 167, 169, 189, 225, 246] [5, 23, 60, 83, 103, 124, 145, 161, 170, 190, 226, 247] [6, 24, 61, 77, 104, 125, 146, 162, 171, 191, 227, 248] [0, 28, 60, 69, 85, 105, 130, 147, 170, 200, 227, 235] [1, 29, 61, 63, 86, 106, 131, 148, 171, 201, 228, 236] [2, 30, 62, 64, 87, 107, 132, 149, 172, 202, 229, 237] [3, 31, 56, 65, 88, 108, 126, 150, 173, 196, 230, 231] [4, 32, 57, 66, 89, 109, 127, 151, 174, 197, 224, 232] [5, 33, 58, 67, 90, 110, 128, 152, 168, 198, 225, 233] [6, 34, 59, 68, 84, 111, 129, 153, 169, 199, 226, 234] [7, 35, 53, 76, 92, 112, 137, 154, 177, 207, 220, 242] [8, 36, 54, 70, 93, 113, 138, 155, 178, 208, 221, 243] [9, 37, 55, 71, 94, 114, 139, 156, 179, 209, 222, 244] [10, 38, 49, 72, 95, 115, 133, 157, 180, 203, 223, 238] [11, 39, 50, 73, 96, 116, 134, 158, 181, 204, 217, 239] [12, 40, 51, 74, 97, 117, 135, 159, 175, 205, 218, 240] [13, 41, 52, 75, 91, 118, 136, 160, 176, 206, 219, 241] [14, 21, 46, 83, 99, 119, 144, 161, 184, 193, 213, 249] [15, 22, 47, 77, 100, 120, 145, 162, 185, 194, 214, 250] [16, 23, 48, 78, 101, 121, 146, 163, 186, 195, 215, 251] [17, 24, 42, 79, 102, 122, 140, 164, 187, 189, 216, 245] [18, 25, 43, 80, 103, 123, 141, 165, 188, 190, 210, 246] [19, 26, 44, 81, 104, 124, 142, 166, 182, 191, 211, 247] [20, 27, 45, 82, 98, 125, 143, 167, 183, 192, 212, 248] [0, 21, 49, 81, 90, 106, 130, 151, 168, 191, 221, 248] [1, 22, 50, 82, 84, 107, 131, 152, 169, 192, 222, 249] [2, 23, 51, 83, 85, 108, 132, 153, 170, 193, 223, 250] [3, 24, 52, 77, 86, 109, 126, 147, 171, 194, 217, 251] [4, 25, 53, 78, 87, 110, 127, 148, 172, 195, 218, 245] [5, 26, 54, 79, 88, 111, 128, 149, 173, 189, 219, 246] [6, 27, 55, 80, 89, 105, 129, 150, 174, 190, 220, 247] [7, 28, 56, 74, 97, 113, 137, 158, 175, 198, 228, 241] [8, 29, 57, 75, 91, 114, 138, 159, 176, 199, 229, 242] [9, 30, 58, 76, 92, 115, 139, 160, 177, 200, 230, 243] [10, 31, 59, 70, 93, 116, 133, 154, 178, 201, 224, 244] [11, 32, 60, 71, 94, 117, 134, 155, 179, 202, 225, 238] [12, 33, 61, 72, 95, 118, 135, 156, 180, 196, 226, 239] [13, 34, 62, 73, 96, 112, 136, 157, 181, 197, 227, 240] [14, 35, 42, 67, 104, 120, 144, 165, 182, 205, 214, 234] [15, 36, 43, 68, 98, 121, 145, 166, 183, 206, 215, 235] [16, 37, 44, 69, 99, 122, 146, 167, 184, 207, 216, 236] [17, 38, 45, 63, 100, 123, 140, 161, 185, 208, 210, 237] [18, 39, 46, 64, 101, 124, 141, 162, 186, 209, 211, 231] [19, 40, 47, 65, 102, 125, 142, 163, 187, 203, 212, 232] [20, 41, 48, 66, 103, 119, 143, 164, 188, 204, 213, 233]
H_Z (63 checks, sparse supports)
[0, 24, 45, 81, 101, 110, 140, 147, 168, 195, 211, 248] [1, 25, 46, 82, 102, 111, 141, 148, 169, 189, 212, 249] [2, 26, 47, 83, 103, 105, 142, 149, 170, 190, 213, 250] [3, 27, 48, 77, 104, 106, 143, 150, 171, 191, 214, 251] [4, 21, 42, 78, 98, 107, 144, 151, 172, 192, 215, 245] [5, 22, 43, 79, 99, 108, 145, 152, 173, 193, 216, 246] [6, 23, 44, 80, 100, 109, 146, 153, 174, 194, 210, 247] [7, 31, 52, 74, 87, 117, 126, 154, 175, 202, 218, 241] [8, 32, 53, 75, 88, 118, 127, 155, 176, 196, 219, 242] [9, 33, 54, 76, 89, 112, 128, 156, 177, 197, 220, 243] [10, 34, 55, 70, 90, 113, 129, 157, 178, 198, 221, 244] [11, 28, 49, 71, 84, 114, 130, 158, 179, 199, 222, 238] [12, 29, 50, 72, 85, 115, 131, 159, 180, 200, 223, 239] [13, 30, 51, 73, 86, 116, 132, 160, 181, 201, 217, 240] [14, 38, 59, 67, 94, 124, 133, 161, 182, 209, 225, 234] [15, 39, 60, 68, 95, 125, 134, 162, 183, 203, 226, 235] [16, 40, 61, 69, 96, 119, 135, 163, 184, 204, 227, 236] [17, 41, 62, 63, 97, 120, 136, 164, 185, 205, 228, 237] [18, 35, 56, 64, 91, 121, 137, 165, 186, 206, 229, 231] [19, 36, 57, 65, 92, 122, 138, 166, 187, 207, 230, 232] [20, 37, 58, 66, 93, 123, 139, 167, 188, 208, 224, 233] [5, 21, 45, 66, 102, 122, 143, 161, 168, 189, 216, 232] [6, 22, 46, 67, 103, 123, 144, 162, 169, 190, 210, 233] [0, 23, 47, 68, 104, 124, 145, 163, 170, 191, 211, 234] [1, 24, 48, 69, 98, 125, 146, 164, 171, 192, 212, 235] [2, 25, 42, 63, 99, 119, 140, 165, 172, 193, 213, 236] [3, 26, 43, 64, 100, 120, 141, 166, 173, 194, 214, 237] [4, 27, 44, 65, 101, 121, 142, 167, 174, 195, 215, 231] [12, 28, 52, 73, 95, 108, 136, 147, 175, 196, 223, 239] [13, 29, 53, 74, 96, 109, 137, 148, 176, 197, 217, 240] [7, 30, 54, 75, 97, 110, 138, 149, 177, 198, 218, 241] [8, 31, 55, 76, 91, 111, 139, 150, 178, 199, 219, 242] [9, 32, 49, 70, 92, 105, 133, 151, 179, 200, 220, 243] [10, 33, 50, 71, 93, 106, 134, 152, 180, 201, 221, 244] [11, 34, 51, 72, 94, 107, 135, 153, 181, 202, 222, 238] [19, 35, 59, 80, 88, 115, 129, 154, 182, 203, 230, 246] [20, 36, 60, 81, 89, 116, 130, 155, 183, 204, 224, 247] [14, 37, 61, 82, 90, 117, 131, 156, 184, 205, 225, 248] [15, 38, 62, 83, 84, 118, 132, 157, 185, 206, 226, 249] [16, 39, 56, 77, 85, 112, 126, 158, 186, 207, 227, 250] [17, 40, 57, 78, 86, 113, 127, 159, 187, 208, 228, 251] [18, 41, 58, 79, 87, 114, 128, 160, 188, 209, 229, 245] [17, 26, 42, 66, 87, 123, 127, 164, 182, 189, 210, 237] [18, 27, 43, 67, 88, 124, 128, 165, 183, 190, 211, 231] [19, 21, 44, 68, 89, 125, 129, 166, 184, 191, 212, 232] [20, 22, 45, 69, 90, 119, 130, 167, 185, 192, 213, 233] [14, 23, 46, 63, 84, 120, 131, 161, 186, 193, 214, 234] [15, 24, 47, 64, 85, 121, 132, 162, 187, 194, 215, 235] [16, 25, 48, 65, 86, 122, 126, 163, 188, 195, 216, 236] [3, 33, 49, 73, 94, 116, 134, 157, 168, 196, 217, 244] [4, 34, 50, 74, 95, 117, 135, 158, 169, 197, 218, 238] [5, 28, 51, 75, 96, 118, 136, 159, 170, 198, 219, 239] [6, 29, 52, 76, 97, 112, 137, 160, 171, 199, 220, 240] [0, 30, 53, 70, 91, 113, 138, 154, 172, 200, 221, 241] [1, 31, 54, 71, 92, 114, 139, 155, 173, 201, 222, 242] [2, 32, 55, 72, 93, 115, 133, 156, 174, 202, 223, 243] [10, 40, 56, 80, 101, 109, 141, 150, 175, 203, 224, 251] [11, 41, 57, 81, 102, 110, 142, 151, 176, 204, 225, 245] [12, 35, 58, 82, 103, 111, 143, 152, 177, 205, 226, 246] [13, 36, 59, 83, 104, 105, 144, 153, 178, 206, 227, 247] [7, 37, 60, 77, 98, 106, 145, 147, 179, 207, 228, 248] [8, 38, 61, 78, 99, 107, 146, 148, 180, 208, 229, 249] [9, 39, 62, 79, 100, 108, 140, 149, 181, 209, 230, 250]
Code ID 252-130-6-b · download JSON · raw on GitHub