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[[378,128,6]] d ≤
n
378
k
128
d
6
kd²/n
12.19
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 6, w_Z = 6 (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)
[59, 66, 120, 127, 183, 364]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[115, 120, 176, 183, 239, 241]
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 12 · H_Z 12 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 6
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×378 (2,2)×1890 (3,2)×9450 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 378 (2,2): 1890 (3,2): 9450 (3,4): 2520
trapping sets H_Z (1,2)×378 (2,2)×1890 (3,2)×9450 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 378 (2,2): 1890 (3,2): 9450 (3,4): 2520

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model Beam 1.0 (claimed, not verified)
date 2026-10-18
family other (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

--- author: @MathysRennela ---

[[378,128,6]]

Direction & hypothesis

Construction I from paper arXiv:2610.19451: fixed circulant lists a,b,c, N=63.

What was searched

Staged candidate validated by the trusted gate (verify/validate_candidate.py, passed: true).

Evidence trail

Distance claim d=6 is an upper bound (witness-backed). Candidate passed the trusted validation gate.

Why it matters

Board-advancing (non-dominated) frontier extension in the ['weight-6', 'unrestricted'] cell.

Reproduction

Candidate document and verdict are in local staging output (not committed).

Parity checks

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