← back to the board
[[520,330,4]] d ≤
n
520
k
330
d
4
kd²/n
10.154
w
12
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · 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 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[0, 1, 19, 39]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[0, 1, 2, 102]
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 1–10 (mean 2.192) · H_Z 1–10 (mean 2.192)
trapping sets H_X (1,1)×50 (2,1)×450 (3,1)×4050 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 50 (1,2): 450 (1,9): 10 (1,10): 10 (2,1): 450 (2,2): 3825 (2,7): 40 (2,8): 100 (2,9): 820 (2,10): 850 (2,16): 20 (2,17): 50 (2,18): 25 (3,1): 4050 (3,2): 32400 (3,3): 1800 (3,4): 9600 (3,5): 60 (3,6): 260 (3,7): 2700 (3,8): 7340 (3,9): 37940 (3,10): 40960 (3,11): 3320 (3,12): 3200 (3,14): 246 (3,15): 890 (3,16): 3580 (3,17): 7312 (3,18): 3851 (3,19): 393 (3,20): 188 (3,23): 60 (3,24): 300 (3,25): 350 (3,26): 100
trapping sets H_Z (1,1)×50 (2,1)×450 (3,1)×4050 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 50 (1,2): 450 (1,9): 10 (1,10): 10 (2,1): 450 (2,2): 3825 (2,7): 40 (2,8): 100 (2,9): 820 (2,10): 850 (2,16): 20 (2,17): 50 (2,18): 25 (3,1): 4050 (3,2): 32400 (3,3): 1800 (3,4): 9600 (3,5): 60 (3,6): 260 (3,7): 2700 (3,8): 7340 (3,9): 37940 (3,10): 40960 (3,11): 3320 (3,12): 3200 (3,14): 242 (3,15): 894 (3,16): 3588 (3,17): 7304 (3,18): 3849 (3,19): 395 (3,20): 188 (3,23): 60 (3,24): 300 (3,25): 350 (3,26): 100

Construction & provenance

authors @hirobumii
provenance submitted through the challenge
novelty novelty not audited
construction G=C20 indexed0..19, multiplication(x,y)=x+y mod20. L_g[t,u]=1 iff u=g+t; R_h[t,u]=1 iff u=t+h. A=[I+L1,I+L3,I+L5,I+L7,I+L9], B=[I+R1,I+R3,I+R5,I+R7,I+R9], binary sums. Twenty-five VV(k,j) blocks indexed5*k+j precede one CC block, each size20. X(j) places A(k) onVV(k,j) and B(j) onCC; Z(k) places B(j)^T onVV(k,j) and A(k)^T onCC. Omit one original row per block: X indices within blocks[0, 2, 3, 4, 6], Z[10, 17, 18, 19, 16]. Keep all other rows,95 per sector. Omitted CC edges partition20 vertices and give degree19.
model GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration) (claimed, not verified)
builds on https://arxiv.org/abs/2111.03654, https://arxiv.org/abs/2607.28795
date 2026-10-17
notes Displayed presentation: maximum check weight 12, maximum qubit degree 19, 190 checks and total incidence 2280.
family lifted product (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

[[520,330,4]] - CSS sparse stabilizer code

Direction & hypothesis

Study sparse finite instances of the construction below and their dimension–distance tradeoff.

What was searched

G=C20 indexed0..19, multiplication(x,y)=x+y mod20. L_g[t,u]=1 iff u=g+t; R_h[t,u]=1 iff u=t+h. A=[I+L1,I+L3,I+L5,I+L7,I+L9], B=[I+R1,I+R3,I+R5,I+R7,I+R9], binary sums. Twenty-five VV(k,j) blocks indexed5*k+j precede one CC block, each size20. X(j) places A(k) onVV(k,j) and B(j) onCC; Z(k) places B(j)^T onVV(k,j) and A(k)^T onCC. Omit one original row per block: X indices within blocks[0, 2, 3, 4, 6], Z[10, 17, 18, 19, 16]. Keep all other rows,95 per sector. Omitted CC edges partition20 vertices and give degree19.

The displayed presentation has maximum check weight 12, maximum qubit degree 19, 190 checks and total incidence 2280.

Evidence trail

The JSON contains weight-4 logical witnesses. Distance confidence: X: upper_bound, Z: upper_bound.

Tools

Recorded workflow configuration: GPT-6 Astra + GPT-6.1 Sol. Official tooling: verify/qldpc_verify.py.

Reproduction

Read codes/520-330-4.json. Allocate zero binary matrices H_X and H_Z with 520 columns. For each row list in checks.X or checks.Z, set the listed zero-based columns to 1, preserving row order. This reconstructs the displayed presentation exactly.

Construction references:

  • https://arxiv.org/abs/2111.03654
  • https://arxiv.org/abs/2607.28795

Local validation: the official verify/validate_candidate.py passed, including independent-seed distance refutation.

Parity checks

X-checks 95 (max weight 12) · Z-checks 95 (max weight 12)
H_X (95 checks, sparse supports)
[1, 2, 101, 104, 201, 206, 301, 308, 401, 410, 501, 502] [2, 3, 102, 105, 202, 207, 302, 309, 402, 411, 502, 503] [3, 4, 103, 106, 203, 208, 303, 310, 403, 412, 503, 504] [4, 5, 104, 107, 204, 209, 304, 311, 404, 413, 504, 505] [5, 6, 105, 108, 205, 210, 305, 312, 405, 414, 505, 506] [6, 7, 106, 109, 206, 211, 306, 313, 406, 415, 506, 507] [7, 8, 107, 110, 207, 212, 307, 314, 407, 416, 507, 508] [8, 9, 108, 111, 208, 213, 308, 315, 408, 417, 508, 509] [9, 10, 109, 112, 209, 214, 309, 316, 409, 418, 509, 510] [10, 11, 110, 113, 210, 215, 310, 317, 410, 419, 510, 511] [11, 12, 111, 114, 211, 216, 311, 318, 400, 411, 511, 512] [12, 13, 112, 115, 212, 217, 312, 319, 401, 412, 512, 513] [13, 14, 113, 116, 213, 218, 300, 313, 402, 413, 513, 514] [14, 15, 114, 117, 214, 219, 301, 314, 403, 414, 514, 515] [15, 16, 115, 118, 200, 215, 302, 315, 404, 415, 515, 516] [16, 17, 116, 119, 201, 216, 303, 316, 405, 416, 516, 517] [17, 18, 100, 117, 202, 217, 304, 317, 406, 417, 517, 518] [18, 19, 101, 118, 203, 218, 305, 318, 407, 418, 518, 519] [0, 19, 102, 119, 204, 219, 306, 319, 408, 419, 500, 519] [20, 21, 120, 123, 220, 225, 320, 327, 420, 429, 500, 503] [21, 22, 121, 124, 221, 226, 321, 328, 421, 430, 501, 504] [23, 24, 123, 126, 223, 228, 323, 330, 423, 432, 503, 506] [24, 25, 124, 127, 224, 229, 324, 331, 424, 433, 504, 507] [25, 26, 125, 128, 225, 230, 325, 332, 425, 434, 505, 508] [26, 27, 126, 129, 226, 231, 326, 333, 426, 435, 506, 509] [27, 28, 127, 130, 227, 232, 327, 334, 427, 436, 507, 510] [28, 29, 128, 131, 228, 233, 328, 335, 428, 437, 508, 511] [29, 30, 129, 132, 229, 234, 329, 336, 429, 438, 509, 512] [30, 31, 130, 133, 230, 235, 330, 337, 430, 439, 510, 513] [31, 32, 131, 134, 231, 236, 331, 338, 420, 431, 511, 514] [32, 33, 132, 135, 232, 237, 332, 339, 421, 432, 512, 515] [33, 34, 133, 136, 233, 238, 320, 333, 422, 433, 513, 516] [34, 35, 134, 137, 234, 239, 321, 334, 423, 434, 514, 517] [35, 36, 135, 138, 220, 235, 322, 335, 424, 435, 515, 518] [36, 37, 136, 139, 221, 236, 323, 336, 425, 436, 516, 519] [37, 38, 120, 137, 222, 237, 324, 337, 426, 437, 500, 517] [38, 39, 121, 138, 223, 238, 325, 338, 427, 438, 501, 518] [20, 39, 122, 139, 224, 239, 326, 339, 428, 439, 502, 519] [40, 41, 140, 143, 240, 245, 340, 347, 440, 449, 500, 505] [41, 42, 141, 144, 241, 246, 341, 348, 441, 450, 501, 506] [42, 43, 142, 145, 242, 247, 342, 349, 442, 451, 502, 507] [44, 45, 144, 147, 244, 249, 344, 351, 444, 453, 504, 509] [45, 46, 145, 148, 245, 250, 345, 352, 445, 454, 505, 510] [46, 47, 146, 149, 246, 251, 346, 353, 446, 455, 506, 511] [47, 48, 147, 150, 247, 252, 347, 354, 447, 456, 507, 512] [48, 49, 148, 151, 248, 253, 348, 355, 448, 457, 508, 513] [49, 50, 149, 152, 249, 254, 349, 356, 449, 458, 509, 514] [50, 51, 150, 153, 250, 255, 350, 357, 450, 459, 510, 515] [51, 52, 151, 154, 251, 256, 351, 358, 440, 451, 511, 516] [52, 53, 152, 155, 252, 257, 352, 359, 441, 452, 512, 517] [53, 54, 153, 156, 253, 258, 340, 353, 442, 453, 513, 518] [54, 55, 154, 157, 254, 259, 341, 354, 443, 454, 514, 519] [55, 56, 155, 158, 240, 255, 342, 355, 444, 455, 500, 515] [56, 57, 156, 159, 241, 256, 343, 356, 445, 456, 501, 516] [57, 58, 140, 157, 242, 257, 344, 357, 446, 457, 502, 517] [58, 59, 141, 158, 243, 258, 345, 358, 447, 458, 503, 518] [40, 59, 142, 159, 244, 259, 346, 359, 448, 459, 504, 519] [60, 61, 160, 163, 260, 265, 360, 367, 460, 469, 500, 507] [61, 62, 161, 164, 261, 266, 361, 368, 461, 470, 501, 508] [62, 63, 162, 165, 262, 267, 362, 369, 462, 471, 502, 509] [63, 64, 163, 166, 263, 268, 363, 370, 463, 472, 503, 510] [65, 66, 165, 168, 265, 270, 365, 372, 465, 474, 505, 512] [66, 67, 166, 169, 266, 271, 366, 373, 466, 475, 506, 513] [67, 68, 167, 170, 267, 272, 367, 374, 467, 476, 507, 514] [68, 69, 168, 171, 268, 273, 368, 375, 468, 477, 508, 515] [69, 70, 169, 172, 269, 274, 369, 376, 469, 478, 509, 516] [70, 71, 170, 173, 270, 275, 370, 377, 470, 479, 510, 517] [71, 72, 171, 174, 271, 276, 371, 378, 460, 471, 511, 518] [72, 73, 172, 175, 272, 277, 372, 379, 461, 472, 512, 519] [73, 74, 173, 176, 273, 278, 360, 373, 462, 473, 500, 513] [74, 75, 174, 177, 274, 279, 361, 374, 463, 474, 501, 514] [75, 76, 175, 178, 260, 275, 362, 375, 464, 475, 502, 515] [76, 77, 176, 179, 261, 276, 363, 376, 465, 476, 503, 516] [77, 78, 160, 177, 262, 277, 364, 377, 466, 477, 504, 517] [78, 79, 161, 178, 263, 278, 365, 378, 467, 478, 505, 518] [60, 79, 162, 179, 264, 279, 366, 379, 468, 479, 506, 519] [80, 81, 180, 183, 280, 285, 380, 387, 480, 489, 500, 509] [81, 82, 181, 184, 281, 286, 381, 388, 481, 490, 501, 510] [82, 83, 182, 185, 282, 287, 382, 389, 482, 491, 502, 511] [83, 84, 183, 186, 283, 288, 383, 390, 483, 492, 503, 512] [84, 85, 184, 187, 284, 289, 384, 391, 484, 493, 504, 513] [85, 86, 185, 188, 285, 290, 385, 392, 485, 494, 505, 514] [87, 88, 187, 190, 287, 292, 387, 394, 487, 496, 507, 516] [88, 89, 188, 191, 288, 293, 388, 395, 488, 497, 508, 517] [89, 90, 189, 192, 289, 294, 389, 396, 489, 498, 509, 518] [90, 91, 190, 193, 290, 295, 390, 397, 490, 499, 510, 519] [91, 92, 191, 194, 291, 296, 391, 398, 480, 491, 500, 511] [92, 93, 192, 195, 292, 297, 392, 399, 481, 492, 501, 512] [93, 94, 193, 196, 293, 298, 380, 393, 482, 493, 502, 513] [94, 95, 194, 197, 294, 299, 381, 394, 483, 494, 503, 514] [95, 96, 195, 198, 280, 295, 382, 395, 484, 495, 504, 515] [96, 97, 196, 199, 281, 296, 383, 396, 485, 496, 505, 516] [97, 98, 180, 197, 282, 297, 384, 397, 486, 497, 506, 517] [98, 99, 181, 198, 283, 298, 385, 398, 487, 498, 507, 518] [80, 99, 182, 199, 284, 299, 386, 399, 488, 499, 508, 519]
H_Z (95 checks, sparse supports)
[0, 19, 20, 37, 40, 55, 60, 73, 80, 91, 500, 519] [0, 1, 21, 38, 41, 56, 61, 74, 81, 92, 500, 501] [1, 2, 22, 39, 42, 57, 62, 75, 82, 93, 501, 502] [2, 3, 20, 23, 43, 58, 63, 76, 83, 94, 502, 503] [3, 4, 21, 24, 44, 59, 64, 77, 84, 95, 503, 504] [4, 5, 22, 25, 40, 45, 65, 78, 85, 96, 504, 505] [5, 6, 23, 26, 41, 46, 66, 79, 86, 97, 505, 506] [6, 7, 24, 27, 42, 47, 60, 67, 87, 98, 506, 507] [7, 8, 25, 28, 43, 48, 61, 68, 88, 99, 507, 508] [8, 9, 26, 29, 44, 49, 62, 69, 80, 89, 508, 509] [10, 11, 28, 31, 46, 51, 64, 71, 82, 91, 510, 511] [11, 12, 29, 32, 47, 52, 65, 72, 83, 92, 511, 512] [12, 13, 30, 33, 48, 53, 66, 73, 84, 93, 512, 513] [13, 14, 31, 34, 49, 54, 67, 74, 85, 94, 513, 514] [14, 15, 32, 35, 50, 55, 68, 75, 86, 95, 514, 515] [15, 16, 33, 36, 51, 56, 69, 76, 87, 96, 515, 516] [16, 17, 34, 37, 52, 57, 70, 77, 88, 97, 516, 517] [17, 18, 35, 38, 53, 58, 71, 78, 89, 98, 517, 518] [18, 19, 36, 39, 54, 59, 72, 79, 90, 99, 518, 519] [100, 119, 120, 137, 140, 155, 160, 173, 180, 191, 500, 517] [100, 101, 121, 138, 141, 156, 161, 174, 181, 192, 501, 518] [101, 102, 122, 139, 142, 157, 162, 175, 182, 193, 502, 519] [102, 103, 120, 123, 143, 158, 163, 176, 183, 194, 500, 503] [103, 104, 121, 124, 144, 159, 164, 177, 184, 195, 501, 504] [104, 105, 122, 125, 140, 145, 165, 178, 185, 196, 502, 505] [105, 106, 123, 126, 141, 146, 166, 179, 186, 197, 503, 506] [106, 107, 124, 127, 142, 147, 160, 167, 187, 198, 504, 507] [107, 108, 125, 128, 143, 148, 161, 168, 188, 199, 505, 508] [108, 109, 126, 129, 144, 149, 162, 169, 180, 189, 506, 509] [109, 110, 127, 130, 145, 150, 163, 170, 181, 190, 507, 510] [110, 111, 128, 131, 146, 151, 164, 171, 182, 191, 508, 511] [111, 112, 129, 132, 147, 152, 165, 172, 183, 192, 509, 512] [112, 113, 130, 133, 148, 153, 166, 173, 184, 193, 510, 513] [113, 114, 131, 134, 149, 154, 167, 174, 185, 194, 511, 514] [114, 115, 132, 135, 150, 155, 168, 175, 186, 195, 512, 515] [115, 116, 133, 136, 151, 156, 169, 176, 187, 196, 513, 516] [117, 118, 135, 138, 153, 158, 171, 178, 189, 198, 515, 518] [118, 119, 136, 139, 154, 159, 172, 179, 190, 199, 516, 519] [200, 219, 220, 237, 240, 255, 260, 273, 280, 291, 500, 515] [200, 201, 221, 238, 241, 256, 261, 274, 281, 292, 501, 516] [201, 202, 222, 239, 242, 257, 262, 275, 282, 293, 502, 517] [202, 203, 220, 223, 243, 258, 263, 276, 283, 294, 503, 518] [203, 204, 221, 224, 244, 259, 264, 277, 284, 295, 504, 519] [204, 205, 222, 225, 240, 245, 265, 278, 285, 296, 500, 505] [205, 206, 223, 226, 241, 246, 266, 279, 286, 297, 501, 506] [206, 207, 224, 227, 242, 247, 260, 267, 287, 298, 502, 507] [207, 208, 225, 228, 243, 248, 261, 268, 288, 299, 503, 508] [208, 209, 226, 229, 244, 249, 262, 269, 280, 289, 504, 509] [209, 210, 227, 230, 245, 250, 263, 270, 281, 290, 505, 510] [210, 211, 228, 231, 246, 251, 264, 271, 282, 291, 506, 511] [211, 212, 229, 232, 247, 252, 265, 272, 283, 292, 507, 512] [212, 213, 230, 233, 248, 253, 266, 273, 284, 293, 508, 513] [213, 214, 231, 234, 249, 254, 267, 274, 285, 294, 509, 514] [214, 215, 232, 235, 250, 255, 268, 275, 286, 295, 510, 515] [215, 216, 233, 236, 251, 256, 269, 276, 287, 296, 511, 516] [216, 217, 234, 237, 252, 257, 270, 277, 288, 297, 512, 517] [218, 219, 236, 239, 254, 259, 272, 279, 290, 299, 514, 519] [300, 319, 320, 337, 340, 355, 360, 373, 380, 391, 500, 513] [300, 301, 321, 338, 341, 356, 361, 374, 381, 392, 501, 514] [301, 302, 322, 339, 342, 357, 362, 375, 382, 393, 502, 515] [302, 303, 320, 323, 343, 358, 363, 376, 383, 394, 503, 516] [303, 304, 321, 324, 344, 359, 364, 377, 384, 395, 504, 517] [304, 305, 322, 325, 340, 345, 365, 378, 385, 396, 505, 518] [305, 306, 323, 326, 341, 346, 366, 379, 386, 397, 506, 519] [306, 307, 324, 327, 342, 347, 360, 367, 387, 398, 500, 507] [307, 308, 325, 328, 343, 348, 361, 368, 388, 399, 501, 508] [308, 309, 326, 329, 344, 349, 362, 369, 380, 389, 502, 509] [309, 310, 327, 330, 345, 350, 363, 370, 381, 390, 503, 510] [310, 311, 328, 331, 346, 351, 364, 371, 382, 391, 504, 511] [311, 312, 329, 332, 347, 352, 365, 372, 383, 392, 505, 512] [312, 313, 330, 333, 348, 353, 366, 373, 384, 393, 506, 513] [313, 314, 331, 334, 349, 354, 367, 374, 385, 394, 507, 514] [314, 315, 332, 335, 350, 355, 368, 375, 386, 395, 508, 515] [315, 316, 333, 336, 351, 356, 369, 376, 387, 396, 509, 516] [316, 317, 334, 337, 352, 357, 370, 377, 388, 397, 510, 517] [317, 318, 335, 338, 353, 358, 371, 378, 389, 398, 511, 518] [400, 419, 420, 437, 440, 455, 460, 473, 480, 491, 500, 511] [400, 401, 421, 438, 441, 456, 461, 474, 481, 492, 501, 512] [401, 402, 422, 439, 442, 457, 462, 475, 482, 493, 502, 513] [402, 403, 420, 423, 443, 458, 463, 476, 483, 494, 503, 514] [403, 404, 421, 424, 444, 459, 464, 477, 484, 495, 504, 515] [404, 405, 422, 425, 440, 445, 465, 478, 485, 496, 505, 516] [405, 406, 423, 426, 441, 446, 466, 479, 486, 497, 506, 517] [406, 407, 424, 427, 442, 447, 460, 467, 487, 498, 507, 518] [407, 408, 425, 428, 443, 448, 461, 468, 488, 499, 508, 519] [408, 409, 426, 429, 444, 449, 462, 469, 480, 489, 500, 509] [409, 410, 427, 430, 445, 450, 463, 470, 481, 490, 501, 510] [410, 411, 428, 431, 446, 451, 464, 471, 482, 491, 502, 511] [411, 412, 429, 432, 447, 452, 465, 472, 483, 492, 503, 512] [412, 413, 430, 433, 448, 453, 466, 473, 484, 493, 504, 513] [413, 414, 431, 434, 449, 454, 467, 474, 485, 494, 505, 514] [414, 415, 432, 435, 450, 455, 468, 475, 486, 495, 506, 515] [416, 417, 434, 437, 452, 457, 470, 477, 488, 497, 508, 517] [417, 418, 435, 438, 453, 458, 471, 478, 489, 498, 509, 518] [418, 419, 436, 439, 454, 459, 472, 479, 490, 499, 510, 519]
Code ID 520-330-4 · download JSON · raw on GitHub