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[[324,166,6]] d ≤
n
324
k
166
d
6
kd²/n
18.444
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)
[45, 75, 85, 130, 161, 181]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[3, 108, 193, 217, 248, 316]
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)×324 (2,2)×162 (3,3)×4977 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 324 (2,2): 162 (2,4): 5022 (3,3): 4977 (3,5): 102591 (3,7): 14706
trapping sets H_Z (1,3)×324 (2,2)×162 (3,3)×4977 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 324 (2,2): 162 (2,4): 5022 (3,3): 4977 (3,5): 102591 (3,7): 14706

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Cornucopia code q=9 (arXiv:2610.19073v2) with column shifts sA=[4,8,6,7,6,1], sB=[4,1,1,6,6,1] 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/324-166-6-b.json: WL-signature match to 324-166-6.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

[[324,166,6]] — Cornucopia q=9 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=9 (sA, sB ∈ Z_9^6 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=[4, 8, 6, 7, 6, 1], sB=[4, 1, 1, 6, 6, 1], 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=9, 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=[4, 8, 6, 7, 6, 1], sB=[4, 1, 1, 6, 6, 1] on Z_9. 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 81 (max weight 12) · Z-checks 81 (max weight 12)
H_X (81 checks, sparse supports)
[13, 53, 60, 88, 114, 136, 166, 190, 226, 267, 276, 298] [14, 45, 61, 89, 115, 137, 167, 191, 227, 268, 277, 299] [15, 46, 62, 81, 116, 138, 168, 192, 228, 269, 278, 300] [16, 47, 54, 82, 108, 139, 169, 193, 229, 261, 270, 301] [17, 48, 55, 83, 109, 140, 170, 194, 230, 262, 271, 302] [9, 49, 56, 84, 110, 141, 162, 195, 231, 263, 272, 303] [10, 50, 57, 85, 111, 142, 163, 196, 232, 264, 273, 304] [11, 51, 58, 86, 112, 143, 164, 197, 233, 265, 274, 305] [12, 52, 59, 87, 113, 135, 165, 189, 225, 266, 275, 297] [22, 44, 69, 97, 123, 145, 175, 199, 235, 258, 285, 307] [23, 36, 70, 98, 124, 146, 176, 200, 236, 259, 286, 308] [24, 37, 71, 90, 125, 147, 177, 201, 237, 260, 287, 309] [25, 38, 63, 91, 117, 148, 178, 202, 238, 252, 279, 310] [26, 39, 64, 92, 118, 149, 179, 203, 239, 253, 280, 311] [18, 40, 65, 93, 119, 150, 171, 204, 240, 254, 281, 312] [19, 41, 66, 94, 120, 151, 172, 205, 241, 255, 282, 313] [20, 42, 67, 95, 121, 152, 173, 206, 242, 256, 283, 314] [21, 43, 68, 96, 122, 144, 174, 198, 234, 257, 284, 306] [4, 35, 78, 106, 132, 154, 184, 208, 217, 249, 294, 316] [5, 27, 79, 107, 133, 155, 185, 209, 218, 250, 295, 317] [6, 28, 80, 99, 134, 156, 186, 210, 219, 251, 296, 318] [7, 29, 72, 100, 126, 157, 187, 211, 220, 243, 288, 319] [8, 30, 73, 101, 127, 158, 188, 212, 221, 244, 289, 320] [0, 31, 74, 102, 128, 159, 180, 213, 222, 245, 290, 321] [1, 32, 75, 103, 129, 160, 181, 214, 223, 246, 291, 322] [2, 33, 76, 104, 130, 161, 182, 215, 224, 247, 292, 323] [3, 34, 77, 105, 131, 153, 183, 207, 216, 248, 293, 315] [1, 40, 80, 87, 115, 141, 163, 193, 217, 253, 294, 303] [2, 41, 72, 88, 116, 142, 164, 194, 218, 254, 295, 304] [3, 42, 73, 89, 108, 143, 165, 195, 219, 255, 296, 305] [4, 43, 74, 81, 109, 135, 166, 196, 220, 256, 288, 297] [5, 44, 75, 82, 110, 136, 167, 197, 221, 257, 289, 298] [6, 36, 76, 83, 111, 137, 168, 189, 222, 258, 290, 299] [7, 37, 77, 84, 112, 138, 169, 190, 223, 259, 291, 300] [8, 38, 78, 85, 113, 139, 170, 191, 224, 260, 292, 301] [0, 39, 79, 86, 114, 140, 162, 192, 216, 252, 293, 302] [10, 49, 71, 96, 124, 150, 172, 202, 226, 262, 285, 312] [11, 50, 63, 97, 125, 151, 173, 203, 227, 263, 286, 313] [12, 51, 64, 98, 117, 152, 174, 204, 228, 264, 287, 314] [13, 52, 65, 90, 118, 144, 175, 205, 229, 265, 279, 306] [14, 53, 66, 91, 119, 145, 176, 206, 230, 266, 280, 307] [15, 45, 67, 92, 120, 146, 177, 198, 231, 267, 281, 308] [16, 46, 68, 93, 121, 147, 178, 199, 232, 268, 282, 309] [17, 47, 69, 94, 122, 148, 179, 200, 233, 269, 283, 310] [9, 48, 70, 95, 123, 149, 171, 201, 225, 261, 284, 311] [19, 31, 62, 105, 133, 159, 181, 211, 235, 244, 276, 321] [20, 32, 54, 106, 134, 160, 182, 212, 236, 245, 277, 322] [21, 33, 55, 107, 126, 161, 183, 213, 237, 246, 278, 323] [22, 34, 56, 99, 127, 153, 184, 214, 238, 247, 270, 315] [23, 35, 57, 100, 128, 154, 185, 215, 239, 248, 271, 316] [24, 27, 58, 101, 129, 155, 186, 207, 240, 249, 272, 317] [25, 28, 59, 102, 130, 156, 187, 208, 241, 250, 273, 318] [26, 29, 60, 103, 131, 157, 188, 209, 242, 251, 274, 319] [18, 30, 61, 104, 132, 158, 180, 210, 234, 243, 275, 320] [6, 28, 67, 107, 114, 142, 168, 190, 220, 244, 280, 321] [7, 29, 68, 99, 115, 143, 169, 191, 221, 245, 281, 322] [8, 30, 69, 100, 116, 135, 170, 192, 222, 246, 282, 323] [0, 31, 70, 101, 108, 136, 162, 193, 223, 247, 283, 315] [1, 32, 71, 102, 109, 137, 163, 194, 224, 248, 284, 316] [2, 33, 63, 103, 110, 138, 164, 195, 216, 249, 285, 317] [3, 34, 64, 104, 111, 139, 165, 196, 217, 250, 286, 318] [4, 35, 65, 105, 112, 140, 166, 197, 218, 251, 287, 319] [5, 27, 66, 106, 113, 141, 167, 189, 219, 243, 279, 320] [15, 37, 76, 98, 123, 151, 177, 199, 229, 253, 289, 312] [16, 38, 77, 90, 124, 152, 178, 200, 230, 254, 290, 313] [17, 39, 78, 91, 125, 144, 179, 201, 231, 255, 291, 314] [9, 40, 79, 92, 117, 145, 171, 202, 232, 256, 292, 306] [10, 41, 80, 93, 118, 146, 172, 203, 233, 257, 293, 307] [11, 42, 72, 94, 119, 147, 173, 204, 225, 258, 294, 308] [12, 43, 73, 95, 120, 148, 174, 205, 226, 259, 295, 309] [13, 44, 74, 96, 121, 149, 175, 206, 227, 260, 296, 310] [14, 36, 75, 97, 122, 150, 176, 198, 228, 252, 288, 311] [24, 46, 58, 89, 132, 160, 186, 208, 238, 262, 271, 303] [25, 47, 59, 81, 133, 161, 187, 209, 239, 263, 272, 304] [26, 48, 60, 82, 134, 153, 188, 210, 240, 264, 273, 305] [18, 49, 61, 83, 126, 154, 180, 211, 241, 265, 274, 297] [19, 50, 62, 84, 127, 155, 181, 212, 242, 266, 275, 298] [20, 51, 54, 85, 128, 156, 182, 213, 234, 267, 276, 299] [21, 52, 55, 86, 129, 157, 183, 214, 235, 268, 277, 300] [22, 53, 56, 87, 130, 158, 184, 215, 236, 269, 278, 301] [23, 45, 57, 88, 131, 159, 185, 207, 237, 261, 270, 302]
H_Z (81 checks, sparse supports)
[5, 35, 57, 102, 134, 143, 185, 197, 219, 245, 273, 316] [6, 27, 58, 103, 126, 135, 186, 189, 220, 246, 274, 317] [7, 28, 59, 104, 127, 136, 187, 190, 221, 247, 275, 318] [8, 29, 60, 105, 128, 137, 188, 191, 222, 248, 276, 319] [0, 30, 61, 106, 129, 138, 180, 192, 223, 249, 277, 320] [1, 31, 62, 107, 130, 139, 181, 193, 224, 250, 278, 321] [2, 32, 54, 99, 131, 140, 182, 194, 216, 251, 270, 322] [3, 33, 55, 100, 132, 141, 183, 195, 217, 243, 271, 323] [4, 34, 56, 101, 133, 142, 184, 196, 218, 244, 272, 315] [14, 44, 66, 93, 116, 152, 167, 206, 228, 254, 282, 307] [15, 36, 67, 94, 108, 144, 168, 198, 229, 255, 283, 308] [16, 37, 68, 95, 109, 145, 169, 199, 230, 256, 284, 309] [17, 38, 69, 96, 110, 146, 170, 200, 231, 257, 285, 310] [9, 39, 70, 97, 111, 147, 162, 201, 232, 258, 286, 311] [10, 40, 71, 98, 112, 148, 163, 202, 233, 259, 287, 312] [11, 41, 63, 90, 113, 149, 164, 203, 225, 260, 279, 313] [12, 42, 64, 91, 114, 150, 165, 204, 226, 252, 280, 314] [13, 43, 65, 92, 115, 151, 166, 205, 227, 253, 281, 306] [23, 53, 75, 84, 125, 161, 176, 215, 237, 263, 291, 298] [24, 45, 76, 85, 117, 153, 177, 207, 238, 264, 292, 299] [25, 46, 77, 86, 118, 154, 178, 208, 239, 265, 293, 300] [26, 47, 78, 87, 119, 155, 179, 209, 240, 266, 294, 301] [18, 48, 79, 88, 120, 156, 171, 210, 241, 267, 295, 302] [19, 49, 80, 89, 121, 157, 172, 211, 242, 268, 296, 303] [20, 50, 72, 81, 122, 158, 173, 212, 234, 269, 288, 304] [21, 51, 73, 82, 123, 159, 174, 213, 235, 261, 289, 305] [22, 52, 74, 83, 124, 160, 175, 214, 236, 262, 290, 297] [8, 32, 62, 84, 129, 161, 181, 212, 224, 246, 272, 300] [0, 33, 54, 85, 130, 153, 182, 213, 216, 247, 273, 301] [1, 34, 55, 86, 131, 154, 183, 214, 217, 248, 274, 302] [2, 35, 56, 87, 132, 155, 184, 215, 218, 249, 275, 303] [3, 27, 57, 88, 133, 156, 185, 207, 219, 250, 276, 304] [4, 28, 58, 89, 134, 157, 186, 208, 220, 251, 277, 305] [5, 29, 59, 81, 126, 158, 187, 209, 221, 243, 278, 297] [6, 30, 60, 82, 127, 159, 188, 210, 222, 244, 270, 298] [7, 31, 61, 83, 128, 160, 180, 211, 223, 245, 271, 299] [17, 41, 71, 93, 120, 143, 172, 194, 233, 255, 281, 309] [9, 42, 63, 94, 121, 135, 173, 195, 225, 256, 282, 310] [10, 43, 64, 95, 122, 136, 174, 196, 226, 257, 283, 311] [11, 44, 65, 96, 123, 137, 175, 197, 227, 258, 284, 312] [12, 36, 66, 97, 124, 138, 176, 189, 228, 259, 285, 313] [13, 37, 67, 98, 125, 139, 177, 190, 229, 260, 286, 314] [14, 38, 68, 90, 117, 140, 178, 191, 230, 252, 287, 306] [15, 39, 69, 91, 118, 141, 179, 192, 231, 253, 279, 307] [16, 40, 70, 92, 119, 142, 171, 193, 232, 254, 280, 308] [26, 50, 80, 102, 111, 152, 163, 203, 242, 264, 290, 318] [18, 51, 72, 103, 112, 144, 164, 204, 234, 265, 291, 319] [19, 52, 73, 104, 113, 145, 165, 205, 235, 266, 292, 320] [20, 53, 74, 105, 114, 146, 166, 206, 236, 267, 293, 321] [21, 45, 75, 106, 115, 147, 167, 198, 237, 268, 294, 322] [22, 46, 76, 107, 116, 148, 168, 199, 238, 269, 295, 323] [23, 47, 77, 99, 108, 149, 169, 200, 239, 261, 296, 315] [24, 48, 78, 100, 109, 150, 170, 201, 240, 262, 288, 316] [25, 49, 79, 101, 110, 151, 162, 202, 241, 263, 289, 317] [26, 35, 59, 89, 111, 156, 165, 208, 239, 251, 273, 299] [18, 27, 60, 81, 112, 157, 166, 209, 240, 243, 274, 300] [19, 28, 61, 82, 113, 158, 167, 210, 241, 244, 275, 301] [20, 29, 62, 83, 114, 159, 168, 211, 242, 245, 276, 302] [21, 30, 54, 84, 115, 160, 169, 212, 234, 246, 277, 303] [22, 31, 55, 85, 116, 161, 170, 213, 235, 247, 278, 304] [23, 32, 56, 86, 108, 153, 162, 214, 236, 248, 270, 305] [24, 33, 57, 87, 109, 154, 163, 215, 237, 249, 271, 297] [25, 34, 58, 88, 110, 155, 164, 207, 238, 250, 272, 298] [8, 44, 68, 98, 120, 147, 174, 199, 221, 260, 282, 308] [0, 36, 69, 90, 121, 148, 175, 200, 222, 252, 283, 309] [1, 37, 70, 91, 122, 149, 176, 201, 223, 253, 284, 310] [2, 38, 71, 92, 123, 150, 177, 202, 224, 254, 285, 311] [3, 39, 63, 93, 124, 151, 178, 203, 216, 255, 286, 312] [4, 40, 64, 94, 125, 152, 179, 204, 217, 256, 287, 313] [5, 41, 65, 95, 117, 144, 171, 205, 218, 257, 279, 314] [6, 42, 66, 96, 118, 145, 172, 206, 219, 258, 280, 306] [7, 43, 67, 97, 119, 146, 173, 198, 220, 259, 281, 307] [17, 53, 77, 107, 129, 138, 183, 190, 230, 269, 291, 317] [9, 45, 78, 99, 130, 139, 184, 191, 231, 261, 292, 318] [10, 46, 79, 100, 131, 140, 185, 192, 232, 262, 293, 319] [11, 47, 80, 101, 132, 141, 186, 193, 233, 263, 294, 320] [12, 48, 72, 102, 133, 142, 187, 194, 225, 264, 295, 321] [13, 49, 73, 103, 134, 143, 188, 195, 226, 265, 296, 322] [14, 50, 74, 104, 126, 135, 180, 196, 227, 266, 288, 323] [15, 51, 75, 105, 127, 136, 181, 197, 228, 267, 289, 315] [16, 52, 76, 106, 128, 137, 182, 189, 229, 268, 290, 316]
Code ID 324-166-6-b · download JSON · raw on GitHub