← back to the board
[[360,16,14]] d ≤
n
360
k
16
d
14
kd²/n
8.711
w
6

Share this result

Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[26, 46, 140, 166, 210, 218, 220, 244, 246, 252, 278, 298, 306, 338]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[91, 101, 123, 133, 135, 157, 167, 177, 247, 251, 257, 319, 325, 343]
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)=[(0, 2), (0, 4), (3, 0)], B(x,y)=[(0, 6), (7, 0), (14, 0)] over F_2[x,y]/(x15-1, y12-1)
model Claude Claude Opus 4.6 + GPT-5.2 + Gemini 3 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

[[360,16,14]] — x/y-swap trinomial BB, MILP-exact d=14

Direction & hypothesis

Bivariate bicycle (BB) codes at (ell,m)=(15,12) targeting the x/y-swap trinomial family (A = x^a + y^b + y^c, B = y^d + x^e + x^f), the only weight-6 trinomial family in our search that reaches d >= 6. This code was found by our LLM-guided evolutionary search described in arXiv:2606.02418 (ensemble Campaign 2-3 run: Claude Opus 4.6 + GPT-5.2 + Gemini 3 Pro Preview mutating a Python program that generates (A, B) exponent-tuple pairs).

What was searched

Population-based evolutionary search (openevolve) over (ell, m) in {(6,6)... (15,12)} at weight-3 trinomial pairs, 500 iterations x 1,000 population across the ensemble; screened via BP-OSD_0 (5,000-10,000 trials) then promoted to exact MILP verification. This code: A(x,y) = y^2 + y^4 + x^3, B(x,y) = y^6 + x^7 + x^14 over F_2[x,y]/(x^15-1, y^12-1).

Evidence trail

BP-OSD_0 gave an upper-bound estimate of d<=16; our MILP-exact solver (evaluation/distance_milp.py, HiGHS/scipy Hamming-weight ILP, one run per logical operator) proved d=14 exactly, checking all 48 logical operators to proven optimality (0 left as unproven incumbents, ~9,350-9,550s wall-clock across the three confirming runs). This is the widest BP-OSD/MILP gap in our own catalog at this size — a useful data point for our paper's broader observation that a heuristic decoder's upper bound and an exact solver's proof can diverge meaningfully even at moderate n. Claim: exact, d=14 (both d_X and d_Z proven at 14 via full MILP).

Dead ends

Uniform (A=B) trinomial pairs at every (ell,m) tried always collapsed to d=2 (proven structurally in our paper, Theorem 1) — this structural degeneracy isn't visible to a decoder-based distance estimate, so any A=B candidate was discarded on sight via the theorem rather than run through the full pipeline. Several nearby x/y-swap trinomials at (15,12) with adjacent exponents gave d<=12 under the same MILP procedure.

Tools

Claude Opus 4.6 + GPT-5.2 + Gemini 3 Pro Preview ensemble (equal selection weight), openevolve harness, evaluation/bb_code.py + evaluation/distance_milp.py from qcode-discovery (see paper for the full pipeline). Apple M4 Max.

Reproduction

Bivariate bicycle at (ell,m)=(15,12): A = {(0,2),(0,4),(3,0)}, B = {(0,6),(7,0),(14,0)} (exponent pairs (x_exp, y_exp)). Construction string matches codes/360-16-14.json.

Parity checks

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