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[[336,20,21]] d ≤
n
336
k
20
d
21
kd²/n
26.25
w
8

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Distance

d_X 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[8, 32, 56, 80, 104, 128, 152, 173, 176, 197, 200, 221, 224, 245, 248, 269, 272, 293, 296, 317, 320]
d_Z 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[6, 21, 30, 45, 54, 69, 78, 93, 102, 117, 126, 141, 150, 165, 174, 198, 222, 246, 270, 294, 318]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @willzeng
provenance submitted through the challenge
novelty novelty not audited
construction Coset two-block group-algebra code (research/kit/coset.build_coset). G = metacyclic(56,6,29) = C56 x| C6, twist r=29 (kit element order: (i,j) -> i*6+j). Non-normal H = <element 3 = (0,3)> (|H|=2, [G:H]=168, |N_G(H)|=168); qubits = 2*[G:H] cosets. H_X=[L(a)|R(b)], H_Z=[R(b)^T|L(a)^T], a=[0, 225, 233, 239] (element indices in G), b=[0, 36, 134, 194] (subset of N_G(H)). Found by simulated annealing with correlated two-element moves; distance = lightest logical witnessed in randomized (RIS) self-refutation (4x190000 gf2_fast trials/side + 0 checkpointed-runner trials/side).
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:2606.17268, arXiv:2306.16400
date 2026-07-14
notes Distance is an upper bound (witnessed). Advances the computed weight-8 unrestricted cell of THIS board; novelty vs literature unverified (arXiv:2606.17268 Appendix F / Table 6 could not be extracted -- manual check required before promotion). Independent adversarial re-check: 3 fresh seeds (940001-940003) x 2M trials/side single-run (gf2_fast), lightest logical 21 each time, nothing lighter (on top of ~2M aggregate trials/side over 10 seeds during the campaign ladder, flat from 8k). Upper bound only. Lit-check 2026-07-14: arXiv:2606.17268 Appendix F Table VI extracted in full and checked - no n=336 entries, max published w8 coset distance is <=20 ([[240,16,<=20]]); Lin-Pryadko enumeration covers only n<=200; novelty vs literature unverified.
family 2BGA coset (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

[[336,20,21]] — coset 2BGA on C56⋊C6 / C2 (weight-8)

Direction & hypothesis

Target: the weight-8 × unrestricted headline ([[336,20,20]], kd²/n 23.81). Hypothesis, learned the hard way (see fieldnote on the simple-group d=2 plateau): record coset codes live on *solvable metacyclic* groups with a large normalizer quotient |N_G(H)/H| — not on simple groups. So: coset 2BGA on C_m⋊C6 with H = C2 at sizes above everything published (n = 336/408/504), annealed with tuned acceptance.

What was searched

5 group configs, 498 anneal restarts, ~173k evaluations (acceptance 5.7–12.1% after raising T_HI 1.2 → 1.8). This code: G = C56⋊C6 (order 336, twist r=29), non-normal H = ⟨(0,3)⟩ ≅ C2, |N_G(H)| = 168, |W| = 84; a = [0,225,233,239], b = [0,36,134,194].

Evidence trail

Ladder [27, 21, 21, 21] — flat from 8k trials; then ~2M aggregate trials/side over 10 independent seeds, all finding 21. Pre-submission: 3 fresh seeds × 2M trials/side single-run, all 21. Claim: upper bound d ≤ 21 — strictly improves [[336,20,20]] (same n, k; d+1).

Dead ends

  • n=168 scale-down (C28⋊C6, |W|=42): 55k evals, zero hits — [[168,20,14]]

    • unbeaten.

  • n=180 two-element correlated moves around the board optimum: 447,859 pair

    • evaluations, zero improvements. Combined with earlier single-move exhaustion, [[180,20,14]] is locally optimal under 1- and 2-element moves; that direction is closed.

  • n=408/504 hits showed classic large-n inflation (63→40, 66→30 still

    • descending at 190k trials); one nominally-flat [[504,16,≤27]] never met the 1M floor and was handed off unpackaged.

  • Lit check: arXiv:2606.17268's full Table VI (Appendix F) was extracted and

    • checked — no n=336 entries; largest published w8 coset distance there is ≤20. (Extraction tip: the HTML renders truncate; read the PDF pages directly.)

Tools

Claude Fable 5 agent campaign; coset builder research/kit/coset.py; gf2_fast; sampler guards from the prior blocked route (odd |b| forces k=0; b = whole normalizer quotient is a k-inflating d=2 degeneracy).

Reproduction

research/kit/coset.py::build_coset with (G, H, a, b) above; construction string in codes/336-20-21.json.

Parity checks

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