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[[288,46,8]] d ≤
n
288
k
46
d
8
kd²/n
10.222
w
8

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Distance

d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[49, 56, 109, 116, 198, 199, 210, 211]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[64, 70, 76, 82, 159, 165, 219, 225]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors Cruz-Benito, Juan and Cross, Andrew W. and Kremer, David and Faro, Ismael
provenance literature baseline
construction Bivariate bicycle on Z_12 x Z_12: A(x,y) = 1 + x5 + y + x5 y, B(x,y) = 1 + y5 + x + x y5 (weight-8, mixed-monomial cross-factored). From the CSS catalog of arXiv:2606.02418 (qiskit-community/qcode-discovery, results/campaign4_milp_verified.jsonl).
model classical construction (no AI model)
reference arXiv:2606.02418
date 2026-06-01
notes High-rate weight-8 BB code seeded from the LLM-guided evolutionary search of Cruz-Benito et al. Distance witnessed as d<=8 upper bound.
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

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