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[[350,72,5]] d ≤
n
350
k
72
d
5
kd²/n
5.143
w
8
X/Z
1.4

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Distance

X/Z asymmetry 1.4 · d_X ≤ 7, d_Z ≤ 5 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[134, 137, 138, 238, 274, 336, 344]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[137, 239, 241, 273, 275]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 7–8 (mean 7.507) · H_Z 7–8 (mean 7.5)
qubit degrees H_X 1–5 (mean 2.96) · H_Z 2–5 (mean 3.0)
trapping sets H_X (1,1)×4 (2,1)×4 (3,1)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 4 (1,2): 142 (1,3): 134 (1,4): 4 (1,5): 66 (2,1): 4 (2,2): 170 (2,3): 818 (2,4): 426 (2,5): 726 (2,6): 968 (2,7): 24 (2,8): 252 (3,1): 4 (3,2): 247 (3,3): 2609 (3,4): 4675 (3,5): 6289 (3,6): 14789 (3,7): 10190 (3,8): 7579 (3,9): 8991 (3,10): 784 (3,11): 2249 (3,12): 8 (3,13): 60
trapping sets H_Z (1,2)×140 (2,2)×140 (3,2)×140 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 140 (1,3): 140 (1,5): 70 (2,2): 140 (2,3): 840 (2,4): 420 (2,5): 700 (2,6): 1050 (2,8): 280 (3,2): 140 (3,3): 2520 (3,4): 4802 (3,5): 5600 (3,6): 15029 (3,7): 11214 (3,8): 7385 (3,9): 9828 (3,10): 560 (3,11): 2590 (3,13): 70

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Check deletion from the board entry [[350,70,10]] (codes/350-70-10.json): rows [90, 134] of H_X removed. Dropping two independent stabiliser generators frees two logical qubits and cannot raise the distance.
model Claude Claude Opus 5 (claimed, not verified)
date 2026-09-21
notes Found by sweeping every single-row deletion of every small board entry and keeping those that land undominated. The parent's remaining checks are unchanged.
family check-deletion (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

[[350,72,5]] by two-row check deletion from 350-70-10

Hypothesis

Every stabiliser generator costs one logical qubit, so dropping two independent generators returns k + 2 on the same n physical qubits at a distance that can only fall.

Whether the trade is worth taking

Usually not. Across 5,033 pair deletions of board entries, 12.5% kept a distance of 3 or more, against 40% for single-row deletion on the same parents. Dropping a second generator costs far more distance than the first. This code is one of the few where it lands undominated anyway, in a k + 2 region of its cell that nothing on the board covers.

Construction

Parent: codes/350-70-10.json. Rows [90, 134] of H_X are removed; every other check is untouched. The pair is checked for independence (the rank must fall by exactly two) and the result for Tanner connectivity before anything else runs.

Distance

Two-sided random information-set search, then the heavier side on its own in the quotient ker(H_opposite) / rowspace(H_own), since a two-sided search returns the minimum over both sides and never surfaces the heavier one. Deleting rows leaves H_X and H_Z with different shapes, so every witness is classified by the conditions it actually satisfies rather than by the search's argument order.

Boundary

Distance is an upper bound from witness search, not a proof. The parent's own distance claim is inherited context. Literature novelty is unverified.

Parity checks

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