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[[420,249,4]] d ≤
n
420
k
249
d
4
kd²/n
9.486
w
12
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · w_X = 12, w_Z = 10 (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, 82]
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 10
qubit degrees H_X 1–8 (mean 2.171) · H_Z 1–10 (mean 2.262)
trapping sets H_X (1,1)×40 (2,1)×360 (3,1)×3240 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 40 (1,2): 360 (1,4): 1 (1,7): 4 (1,8): 15 (2,1): 360 (2,2): 3060 (2,3): 4 (2,4): 36 (2,5): 12 (2,6): 76 (2,7): 300 (2,8): 1020 (2,10): 4 (2,13): 28 (2,14): 44 (3,1): 3240 (3,2): 25926 (3,3): 1560 (3,4): 8616 (3,5): 756 (3,6): 4242 (3,7): 12624 (3,8): 40146 (3,9): 1428 (3,10): 4212 (3,11): 292 (3,12): 788 (3,13): 3408 (3,14): 5296 (3,15): 232 (3,16): 366 (3,18): 33 (3,19): 214 (3,20): 229
trapping sets H_Z (1,1)×40 (2,1)×280 (3,1)×1960 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 40 (1,2): 360 (1,5): 1 (1,9): 5 (1,10): 14 (2,1): 280 (2,2): 2380 (2,4): 4 (2,5): 36 (2,7): 16 (2,8): 76 (2,9): 364 (2,10): 952 (2,13): 5 (2,17): 45 (2,18): 45 (3,1): 1960 (3,2): 15680 (3,3): 846 (3,4): 4596 (3,5): 788 (3,6): 160 (3,7): 1108 (3,8): 4386 (3,9): 14544 (3,10): 36262 (3,11): 1288 (3,12): 2836 (3,13): 460 (3,14): 4 (3,15): 544 (3,16): 960 (3,17): 5284 (3,18): 5388 (3,19): 288 (3,20): 272 (3,21): 55 (3,24): 90 (3,25): 405 (3,26): 270

Construction & provenance

authors @hirobumii
provenance submitted through the challenge
novelty novelty not audited
construction G=C20, indices0..19, product 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], binary sums. Twenty VV(k,j) blocks indexed4*k+j precede one CC block, each size20. X(j) places A(k) on VV(k,j) and B(j) on CC; Z(k) places B(j)^T on VV(k,j) and A(k)^T on CC. Keep first independent original rows,76 X and95 Z.
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 18, 171 checks and total incidence 1862.
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

[[420,249,4]] - CSS sparse stabilizer code

Direction & hypothesis

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

What was searched

G=C20, indices0..19, product 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], binary sums. Twenty VV(k,j) blocks indexed4*k+j precede one CC block, each size20. X(j) places A(k) on VV(k,j) and B(j) on CC; Z(k) places B(j)^T on VV(k,j) and A(k)^T on CC. Keep first independent original rows,76 X and95 Z.

The displayed presentation has maximum check weight 12, maximum qubit degree 18, 171 checks and total incidence 1862.

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