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[[396,202,6]] d ≤
n
396
k
202
d
6
kd²/n
18.364
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)
[22, 108, 137, 148, 256, 317]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[104, 155, 187, 273, 302, 334]
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)×396 (2,2)×363 (3,3)×8910 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 396 (2,2): 363 (2,4): 5808 (3,3): 8910 (3,5): 116314 (3,7): 15268
trapping sets H_Z (1,3)×396 (2,2)×363 (3,3)×8910 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 396 (2,2): 363 (2,4): 5808 (3,3): 8910 (3,5): 116314 (3,7): 15268

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Cornucopia code q=11 (arXiv:2610.19073v2) with column shifts sA=[3,0,7,9,2,4], sB=[4,0,2,6,8,5] 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/396-202-6-b.json: WL-signature match to 396-202-6.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

[[396,202,6]] — Cornucopia q=11 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=11 (sA, sB ∈ Z_11^6 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=[3, 0, 7, 9, 2, 4], sB=[4, 0, 2, 6, 8, 5], 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=11, 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=[3, 0, 7, 9, 2, 4], sB=[4, 0, 2, 6, 8, 5] on Z_11. 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 99 (max weight 12) · Z-checks 99 (max weight 12)
H_X (99 checks, sparse supports)
[14, 55, 73, 108, 134, 169, 202, 231, 277, 325, 338, 368] [15, 56, 74, 109, 135, 170, 203, 232, 278, 326, 339, 369] [16, 57, 75, 99, 136, 171, 204, 233, 279, 327, 340, 370] [17, 58, 76, 100, 137, 172, 205, 234, 280, 328, 330, 371] [18, 59, 66, 101, 138, 173, 206, 235, 281, 329, 331, 372] [19, 60, 67, 102, 139, 174, 207, 236, 282, 319, 332, 373] [20, 61, 68, 103, 140, 175, 208, 237, 283, 320, 333, 363] [21, 62, 69, 104, 141, 165, 198, 238, 284, 321, 334, 364] [11, 63, 70, 105, 142, 166, 199, 239, 285, 322, 335, 365] [12, 64, 71, 106, 132, 167, 200, 240, 275, 323, 336, 366] [13, 65, 72, 107, 133, 168, 201, 241, 276, 324, 337, 367] [25, 44, 84, 119, 145, 180, 213, 242, 288, 314, 349, 379] [26, 45, 85, 120, 146, 181, 214, 243, 289, 315, 350, 380] [27, 46, 86, 110, 147, 182, 215, 244, 290, 316, 351, 381] [28, 47, 87, 111, 148, 183, 216, 245, 291, 317, 341, 382] [29, 48, 77, 112, 149, 184, 217, 246, 292, 318, 342, 383] [30, 49, 78, 113, 150, 185, 218, 247, 293, 308, 343, 384] [31, 50, 79, 114, 151, 186, 219, 248, 294, 309, 344, 374] [32, 51, 80, 115, 152, 176, 209, 249, 295, 310, 345, 375] [22, 52, 81, 116, 153, 177, 210, 250, 296, 311, 346, 376] [23, 53, 82, 117, 143, 178, 211, 251, 286, 312, 347, 377] [24, 54, 83, 118, 144, 179, 212, 252, 287, 313, 348, 378] [3, 33, 95, 130, 156, 191, 224, 253, 266, 303, 360, 390] [4, 34, 96, 131, 157, 192, 225, 254, 267, 304, 361, 391] [5, 35, 97, 121, 158, 193, 226, 255, 268, 305, 362, 392] [6, 36, 98, 122, 159, 194, 227, 256, 269, 306, 352, 393] [7, 37, 88, 123, 160, 195, 228, 257, 270, 307, 353, 394] [8, 38, 89, 124, 161, 196, 229, 258, 271, 297, 354, 395] [9, 39, 90, 125, 162, 197, 230, 259, 272, 298, 355, 385] [10, 40, 91, 126, 163, 187, 220, 260, 273, 299, 356, 386] [0, 41, 92, 127, 164, 188, 221, 261, 274, 300, 357, 387] [1, 42, 93, 128, 154, 189, 222, 262, 264, 301, 358, 388] [2, 43, 94, 129, 155, 190, 223, 263, 265, 302, 359, 389] [4, 47, 88, 106, 141, 167, 203, 235, 264, 310, 358, 371] [5, 48, 89, 107, 142, 168, 204, 236, 265, 311, 359, 372] [6, 49, 90, 108, 132, 169, 205, 237, 266, 312, 360, 373] [7, 50, 91, 109, 133, 170, 206, 238, 267, 313, 361, 363] [8, 51, 92, 99, 134, 171, 207, 239, 268, 314, 362, 364] [9, 52, 93, 100, 135, 172, 208, 240, 269, 315, 352, 365] [10, 53, 94, 101, 136, 173, 198, 241, 270, 316, 353, 366] [0, 54, 95, 102, 137, 174, 199, 231, 271, 317, 354, 367] [1, 44, 96, 103, 138, 175, 200, 232, 272, 318, 355, 368] [2, 45, 97, 104, 139, 165, 201, 233, 273, 308, 356, 369] [3, 46, 98, 105, 140, 166, 202, 234, 274, 309, 357, 370] [15, 58, 77, 117, 152, 178, 214, 246, 275, 321, 347, 382] [16, 59, 78, 118, 153, 179, 215, 247, 276, 322, 348, 383] [17, 60, 79, 119, 143, 180, 216, 248, 277, 323, 349, 384] [18, 61, 80, 120, 144, 181, 217, 249, 278, 324, 350, 374] [19, 62, 81, 110, 145, 182, 218, 250, 279, 325, 351, 375] [20, 63, 82, 111, 146, 183, 219, 251, 280, 326, 341, 376] [21, 64, 83, 112, 147, 184, 209, 252, 281, 327, 342, 377] [11, 65, 84, 113, 148, 185, 210, 242, 282, 328, 343, 378] [12, 55, 85, 114, 149, 186, 211, 243, 283, 329, 344, 379] [13, 56, 86, 115, 150, 176, 212, 244, 284, 319, 345, 380] [14, 57, 87, 116, 151, 177, 213, 245, 285, 320, 346, 381] [26, 36, 66, 128, 163, 189, 225, 257, 286, 299, 336, 393] [27, 37, 67, 129, 164, 190, 226, 258, 287, 300, 337, 394] [28, 38, 68, 130, 154, 191, 227, 259, 288, 301, 338, 395] [29, 39, 69, 131, 155, 192, 228, 260, 289, 302, 339, 385] [30, 40, 70, 121, 156, 193, 229, 261, 290, 303, 340, 386] [31, 41, 71, 122, 157, 194, 230, 262, 291, 304, 330, 387] [32, 42, 72, 123, 158, 195, 220, 263, 292, 305, 331, 388] [22, 43, 73, 124, 159, 196, 221, 253, 293, 306, 332, 389] [23, 33, 74, 125, 160, 197, 222, 254, 294, 307, 333, 390] [24, 34, 75, 126, 161, 187, 223, 255, 295, 297, 334, 391] [25, 35, 76, 127, 162, 188, 224, 256, 296, 298, 335, 392] [2, 37, 80, 121, 139, 174, 206, 236, 268, 297, 343, 391] [3, 38, 81, 122, 140, 175, 207, 237, 269, 298, 344, 392] [4, 39, 82, 123, 141, 165, 208, 238, 270, 299, 345, 393] [5, 40, 83, 124, 142, 166, 198, 239, 271, 300, 346, 394] [6, 41, 84, 125, 132, 167, 199, 240, 272, 301, 347, 395] [7, 42, 85, 126, 133, 168, 200, 241, 273, 302, 348, 385] [8, 43, 86, 127, 134, 169, 201, 231, 274, 303, 349, 386] [9, 33, 87, 128, 135, 170, 202, 232, 264, 304, 350, 387] [10, 34, 77, 129, 136, 171, 203, 233, 265, 305, 351, 388] [0, 35, 78, 130, 137, 172, 204, 234, 266, 306, 341, 389] [1, 36, 79, 131, 138, 173, 205, 235, 267, 307, 342, 390] [13, 48, 91, 110, 150, 185, 217, 247, 279, 308, 354, 380] [14, 49, 92, 111, 151, 186, 218, 248, 280, 309, 355, 381] [15, 50, 93, 112, 152, 176, 219, 249, 281, 310, 356, 382] [16, 51, 94, 113, 153, 177, 209, 250, 282, 311, 357, 383] [17, 52, 95, 114, 143, 178, 210, 251, 283, 312, 358, 384] [18, 53, 96, 115, 144, 179, 211, 252, 284, 313, 359, 374] [19, 54, 97, 116, 145, 180, 212, 242, 285, 314, 360, 375] [20, 44, 98, 117, 146, 181, 213, 243, 275, 315, 361, 376] [21, 45, 88, 118, 147, 182, 214, 244, 276, 316, 362, 377] [11, 46, 89, 119, 148, 183, 215, 245, 277, 317, 352, 378] [12, 47, 90, 120, 149, 184, 216, 246, 278, 318, 353, 379] [24, 59, 69, 99, 161, 196, 228, 258, 290, 319, 332, 369] [25, 60, 70, 100, 162, 197, 229, 259, 291, 320, 333, 370] [26, 61, 71, 101, 163, 187, 230, 260, 292, 321, 334, 371] [27, 62, 72, 102, 164, 188, 220, 261, 293, 322, 335, 372] [28, 63, 73, 103, 154, 189, 221, 262, 294, 323, 336, 373] [29, 64, 74, 104, 155, 190, 222, 263, 295, 324, 337, 363] [30, 65, 75, 105, 156, 191, 223, 253, 296, 325, 338, 364] [31, 55, 76, 106, 157, 192, 224, 254, 286, 326, 339, 365] [32, 56, 66, 107, 158, 193, 225, 255, 287, 327, 340, 366] [22, 57, 67, 108, 159, 194, 226, 256, 288, 328, 330, 367] [23, 58, 68, 109, 160, 195, 227, 257, 289, 329, 331, 368]
H_Z (99 checks, sparse supports)
[7, 39, 69, 126, 163, 165, 228, 238, 273, 299, 334, 385] [8, 40, 70, 127, 164, 166, 229, 239, 274, 300, 335, 386] [9, 41, 71, 128, 154, 167, 230, 240, 264, 301, 336, 387] [10, 42, 72, 129, 155, 168, 220, 241, 265, 302, 337, 388] [0, 43, 73, 130, 156, 169, 221, 231, 266, 303, 338, 389] [1, 33, 74, 131, 157, 170, 222, 232, 267, 304, 339, 390] [2, 34, 75, 121, 158, 171, 223, 233, 268, 305, 340, 391] [3, 35, 76, 122, 159, 172, 224, 234, 269, 306, 330, 392] [4, 36, 66, 123, 160, 173, 225, 235, 270, 307, 331, 393] [5, 37, 67, 124, 161, 174, 226, 236, 271, 297, 332, 394] [6, 38, 68, 125, 162, 175, 227, 237, 272, 298, 333, 395] [18, 50, 80, 115, 141, 176, 206, 249, 284, 310, 345, 374] [19, 51, 81, 116, 142, 177, 207, 250, 285, 311, 346, 375] [20, 52, 82, 117, 132, 178, 208, 251, 275, 312, 347, 376] [21, 53, 83, 118, 133, 179, 198, 252, 276, 313, 348, 377] [11, 54, 84, 119, 134, 180, 199, 242, 277, 314, 349, 378] [12, 44, 85, 120, 135, 181, 200, 243, 278, 315, 350, 379] [13, 45, 86, 110, 136, 182, 201, 244, 279, 316, 351, 380] [14, 46, 87, 111, 137, 183, 202, 245, 280, 317, 341, 381] [15, 47, 77, 112, 138, 184, 203, 246, 281, 318, 342, 382] [16, 48, 78, 113, 139, 185, 204, 247, 282, 308, 343, 383] [17, 49, 79, 114, 140, 186, 205, 248, 283, 309, 344, 384] [29, 61, 91, 104, 152, 187, 217, 260, 295, 321, 356, 363] [30, 62, 92, 105, 153, 188, 218, 261, 296, 322, 357, 364] [31, 63, 93, 106, 143, 189, 219, 262, 286, 323, 358, 365] [32, 64, 94, 107, 144, 190, 209, 263, 287, 324, 359, 366] [22, 65, 95, 108, 145, 191, 210, 253, 288, 325, 360, 367] [23, 55, 96, 109, 146, 192, 211, 254, 289, 326, 361, 368] [24, 56, 97, 99, 147, 193, 212, 255, 290, 327, 362, 369] [25, 57, 98, 100, 148, 194, 213, 256, 291, 328, 352, 370] [26, 58, 88, 101, 149, 195, 214, 257, 292, 329, 353, 371] [27, 59, 89, 102, 150, 196, 215, 258, 293, 319, 354, 372] [28, 60, 90, 103, 151, 197, 216, 259, 294, 320, 355, 373] [0, 40, 72, 102, 159, 196, 220, 261, 271, 306, 332, 367] [1, 41, 73, 103, 160, 197, 221, 262, 272, 307, 333, 368] [2, 42, 74, 104, 161, 187, 222, 263, 273, 297, 334, 369] [3, 43, 75, 105, 162, 188, 223, 253, 274, 298, 335, 370] [4, 33, 76, 106, 163, 189, 224, 254, 264, 299, 336, 371] [5, 34, 66, 107, 164, 190, 225, 255, 265, 300, 337, 372] [6, 35, 67, 108, 154, 191, 226, 256, 266, 301, 338, 373] [7, 36, 68, 109, 155, 192, 227, 257, 267, 302, 339, 363] [8, 37, 69, 99, 156, 193, 228, 258, 268, 303, 340, 364] [9, 38, 70, 100, 157, 194, 229, 259, 269, 304, 330, 365] [10, 39, 71, 101, 158, 195, 230, 260, 270, 305, 331, 366] [11, 51, 83, 113, 148, 174, 209, 239, 282, 317, 343, 378] [12, 52, 84, 114, 149, 175, 210, 240, 283, 318, 344, 379] [13, 53, 85, 115, 150, 165, 211, 241, 284, 308, 345, 380] [14, 54, 86, 116, 151, 166, 212, 231, 285, 309, 346, 381] [15, 44, 87, 117, 152, 167, 213, 232, 275, 310, 347, 382] [16, 45, 77, 118, 153, 168, 214, 233, 276, 311, 348, 383] [17, 46, 78, 119, 143, 169, 215, 234, 277, 312, 349, 384] [18, 47, 79, 120, 144, 170, 216, 235, 278, 313, 350, 374] [19, 48, 80, 110, 145, 171, 217, 236, 279, 314, 351, 375] [20, 49, 81, 111, 146, 172, 218, 237, 280, 315, 341, 376] [21, 50, 82, 112, 147, 173, 219, 238, 281, 316, 342, 377] [22, 62, 94, 124, 137, 185, 198, 250, 293, 328, 354, 389] [23, 63, 95, 125, 138, 186, 199, 251, 294, 329, 355, 390] [24, 64, 96, 126, 139, 176, 200, 252, 295, 319, 356, 391] [25, 65, 97, 127, 140, 177, 201, 242, 296, 320, 357, 392] [26, 55, 98, 128, 141, 178, 202, 243, 286, 321, 358, 393] [27, 56, 88, 129, 142, 179, 203, 244, 287, 322, 359, 394] [28, 57, 89, 130, 132, 180, 204, 245, 288, 323, 360, 395] [29, 58, 90, 131, 133, 181, 205, 246, 289, 324, 361, 385] [30, 59, 91, 121, 134, 182, 206, 247, 290, 325, 362, 386] [31, 60, 92, 122, 135, 183, 207, 248, 291, 326, 352, 387] [32, 61, 93, 123, 136, 184, 208, 249, 292, 327, 353, 388] [31, 33, 73, 105, 135, 192, 202, 253, 294, 304, 339, 365] [32, 34, 74, 106, 136, 193, 203, 254, 295, 305, 340, 366] [22, 35, 75, 107, 137, 194, 204, 255, 296, 306, 330, 367] [23, 36, 76, 108, 138, 195, 205, 256, 286, 307, 331, 368] [24, 37, 66, 109, 139, 196, 206, 257, 287, 297, 332, 369] [25, 38, 67, 99, 140, 197, 207, 258, 288, 298, 333, 370] [26, 39, 68, 100, 141, 187, 208, 259, 289, 299, 334, 371] [27, 40, 69, 101, 142, 188, 198, 260, 290, 300, 335, 372] [28, 41, 70, 102, 132, 189, 199, 261, 291, 301, 336, 373] [29, 42, 71, 103, 133, 190, 200, 262, 292, 302, 337, 363] [30, 43, 72, 104, 134, 191, 201, 263, 293, 303, 338, 364] [9, 44, 84, 116, 146, 181, 213, 242, 272, 315, 350, 376] [10, 45, 85, 117, 147, 182, 214, 243, 273, 316, 351, 377] [0, 46, 86, 118, 148, 183, 215, 244, 274, 317, 341, 378] [1, 47, 87, 119, 149, 184, 216, 245, 264, 318, 342, 379] [2, 48, 77, 120, 150, 185, 217, 246, 265, 308, 343, 380] [3, 49, 78, 110, 151, 186, 218, 247, 266, 309, 344, 381] [4, 50, 79, 111, 152, 176, 219, 248, 267, 310, 345, 382] [5, 51, 80, 112, 153, 177, 209, 249, 268, 311, 346, 383] [6, 52, 81, 113, 143, 178, 210, 250, 269, 312, 347, 384] [7, 53, 82, 114, 144, 179, 211, 251, 270, 313, 348, 374] [8, 54, 83, 115, 145, 180, 212, 252, 271, 314, 349, 375] [20, 55, 95, 127, 157, 170, 224, 231, 283, 326, 361, 387] [21, 56, 96, 128, 158, 171, 225, 232, 284, 327, 362, 388] [11, 57, 97, 129, 159, 172, 226, 233, 285, 328, 352, 389] [12, 58, 98, 130, 160, 173, 227, 234, 275, 329, 353, 390] [13, 59, 88, 131, 161, 174, 228, 235, 276, 319, 354, 391] [14, 60, 89, 121, 162, 175, 229, 236, 277, 320, 355, 392] [15, 61, 90, 122, 163, 165, 230, 237, 278, 321, 356, 393] [16, 62, 91, 123, 164, 166, 220, 238, 279, 322, 357, 394] [17, 63, 92, 124, 154, 167, 221, 239, 280, 323, 358, 395] [18, 64, 93, 125, 155, 168, 222, 240, 281, 324, 359, 385] [19, 65, 94, 126, 156, 169, 223, 241, 282, 325, 360, 386]
Code ID 396-202-6-b · download JSON · raw on GitHub