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[[360,32,6]] d ≤
n
360
k
32
d
6
kd²/n
3.2
w
8

Share this result

Distance

d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[180, 182, 184, 186, 188, 190]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[133, 135, 137, 139, 141, 143]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @cbjuan
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle (A=[(0,0),(4,0),(0,2),(4,2)], B=[(0,0),(8,0),(0,4),(2,4)]) on (ell,m)=(15,12)
model Claude Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview (claimed, not verified)
date 2026-08-05
family bivariate bicycle (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

[[360,32,6]] — weight-8 mixed-monomial BB, MILP-exact d=6

Direction & hypothesis

Campaign 4's mixed-monomial ansatz also explored 4-term A/B pairs (weight-8 checks rather than weight-6), trading check locality for reach on k at fixed n. This code sits at that weight-8 corner of the (15,12) lattice already explored for weight-6 in [notes/360-16-14.md](360-16-14.md). Found by our LLM-guided evolutionary search (arXiv:2606.02418); this specific representation appears only in our raw Campaign 4 result log, not in the paper's hand-curated appendix table, so we attribute it to the Campaign 4 ensemble by source-file provenance (Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview) rather than a specific cited row.

What was searched

300 iterations x 750 population, mixed-monomial ansatz allowing 4-term A/B pairs at (ell,m)=(15,12). This code: A(x,y) = 1 + x^4 + y^2 + x^4y^2, B(x,y) = 1 + x^8 + y^4 + x^2y^4 over F_2[x,y]/(x^15-1, y^12-1) — each polynomial factors as a product of two binomials (A = (1+x^4)(1+y^2), B = (1+x^8)(1+y^4), up to the x^2y^4 cross term), a structured 4-term form distinct from the single-term-per-axis trinomials elsewhere in this batch.

Evidence trail

MILP-exact (evaluation/distance_milp.py) proved d=6 exactly: all 64 logical operators checked, 0 unproven incumbents, ~1,743s wall-clock. Claim: exact, d=6.

Dead ends

The clean product structure A=(1+x^4)(1+y^2) generalizes to other divisor pairs of (15,12); most nearby divisor choices we logged either reduced to d=2 (an A-B relationship close enough to the proven A=B trap) or gave lower k at the same d, and none exceeded d=6 at this k in the resulting sweep.

Tools

Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview ensemble, openevolve harness, evaluation/bb_code.py + evaluation/distance_milp.py from qcode-discovery. 64-core server.

Reproduction

Bivariate bicycle at (ell,m)=(15,12): A = {(0,0),(4,0),(0,2),(4,2)}, B = {(0,0),(8,0),(0,4),(2,4)}. Construction string matches codes/360-32-6.json.

Parity checks

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