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[[360,24,10]] d ≤
n
360
k
24
d
10
kd²/n
6.667
w
6

Share this result

Distance

d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[16, 49, 66, 122, 229, 302, 319, 320, 336, 359]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[15, 16, 33, 49, 50, 67, 139, 247, 319, 359]
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(x,y)=[(3, 0), (0, 1), (0, 2)], B(x,y)=[(21, 0), (0, 1), (0, 2)] over F_2[x,y]/(x30-1, y6-1)
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 ≤ 6 (computed)

How this code was found

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

[[360,24,10]] — x/y-swap trinomial BB, MILP-exact d=10

Direction & hypothesis

The mixed-monomial ansatz (Campaign 4) is a superset of the weight-3 x/y-swap trinomial family — a 4-6 term budget with all-but-3 coefficients pinned to zero degenerates to the trinomial case. This code is exactly that: a trinomial pair the ansatz search rediscovered at a larger lattice, (ell,m)=(30,6). Found by our LLM-guided evolutionary search (arXiv:2606.02418, Campaign 4 ensemble: Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview).

What was searched

300 iterations x 750 population, mixed-monomial ansatz swept up to (ell,m)=(30,6). This code: A(x,y) = x^3 + y + y^2, B(x,y) = x^21 + y + y^2 — same pure-x/pure-y trinomial shape as [[144,24,6]] and [[360,16,14]] (A and B share the y+y^2 term, differ only in the x^3 vs x^21 term), over F_2[x,y]/(x^30-1, y^6-1).

Evidence trail

MILP-exact (evaluation/distance_milp.py) proved d=10 exactly: all 48 logical operators checked to proven optimality, 0 unproven incumbents, ~2,846s wall-clock (the longest-running exact certification in this batch — consistent with our paper's note that exact MILP scales worse than BP-OSD as n grows). A duplicate representation of this (n,k,d) triple in our raw campaign log stopped at milp_incumbent (feasible but not proven optimal) before a later run completed the proof; we submit only the fully-proven representation. Claim: exact, d=10.

Dead ends

Trinomial pairs sharing A and B's y-part but with other x-exponent choices at (30,6) mostly gave d<=8 under MILP; this required the specific x^3/x^21 split (21 = -9 mod 30, i.e. a specific divisor relationship on Z_30) to reach d=10.

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)=(30,6): A = {(3,0),(0,1),(0,2)}, B = {(21,0),(0,1),(0,2)}. Construction string matches codes/360-24-10.json.

Parity checks

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