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[[288,48,8]] d ≤
n
288
k
48
d
8
kd²/n
10.667
w
10

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Distance

d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[24, 28, 60, 64, 102, 106, 138, 142]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[3, 6, 24, 33, 72, 81, 99, 102]
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 + x2 + x10 + y3 + y9, B(x,y) = 1 + x3 + x9 + y2 + y10 (weight-10, mixed-monomial). From the CSS catalog of arXiv:2606.02418 (qiskit-community/qcode-discovery).
model classical construction (no AI model)
reference arXiv:2606.02418
date 2026-06-01
notes High-rate weight-10 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, 3, 9, 24, 120, 144, 146, 154, 180, 252] [1, 4, 10, 25, 121, 145, 147, 155, 181, 253] [2, 5, 11, 26, 122, 144, 146, 148, 182, 254] [0, 3, 6, 27, 123, 145, 147, 149, 183, 255] [1, 4, 7, 28, 124, 146, 148, 150, 184, 256] [2, 5, 8, 29, 125, 147, 149, 151, 185, 257] [3, 6, 9, 30, 126, 148, 150, 152, 186, 258] [4, 7, 10, 31, 127, 149, 151, 153, 187, 259] [5, 8, 11, 32, 128, 150, 152, 154, 188, 260] [0, 6, 9, 33, 129, 151, 153, 155, 189, 261] [1, 7, 10, 34, 130, 144, 152, 154, 190, 262] [2, 8, 11, 35, 131, 145, 153, 155, 191, 263] [12, 15, 21, 36, 132, 156, 158, 166, 192, 264] [13, 16, 22, 37, 133, 157, 159, 167, 193, 265] [14, 17, 23, 38, 134, 156, 158, 160, 194, 266] [12, 15, 18, 39, 135, 157, 159, 161, 195, 267] [13, 16, 19, 40, 136, 158, 160, 162, 196, 268] [14, 17, 20, 41, 137, 159, 161, 163, 197, 269] [15, 18, 21, 42, 138, 160, 162, 164, 198, 270] [16, 19, 22, 43, 139, 161, 163, 165, 199, 271] [17, 20, 23, 44, 140, 162, 164, 166, 200, 272] [12, 18, 21, 45, 141, 163, 165, 167, 201, 273] [13, 19, 22, 46, 142, 156, 164, 166, 202, 274] [14, 20, 23, 47, 143, 157, 165, 167, 203, 275] [0, 24, 27, 33, 48, 168, 170, 178, 204, 276] [1, 25, 28, 34, 49, 169, 171, 179, 205, 277] [2, 26, 29, 35, 50, 168, 170, 172, 206, 278] [3, 24, 27, 30, 51, 169, 171, 173, 207, 279] [4, 25, 28, 31, 52, 170, 172, 174, 208, 280] [5, 26, 29, 32, 53, 171, 173, 175, 209, 281] [6, 27, 30, 33, 54, 172, 174, 176, 210, 282] [7, 28, 31, 34, 55, 173, 175, 177, 211, 283] [8, 29, 32, 35, 56, 174, 176, 178, 212, 284] [9, 24, 30, 33, 57, 175, 177, 179, 213, 285] [10, 25, 31, 34, 58, 168, 176, 178, 214, 286] [11, 26, 32, 35, 59, 169, 177, 179, 215, 287] [12, 36, 39, 45, 60, 144, 180, 182, 190, 216] [13, 37, 40, 46, 61, 145, 181, 183, 191, 217] [14, 38, 41, 47, 62, 146, 180, 182, 184, 218] [15, 36, 39, 42, 63, 147, 181, 183, 185, 219] [16, 37, 40, 43, 64, 148, 182, 184, 186, 220] [17, 38, 41, 44, 65, 149, 183, 185, 187, 221] [18, 39, 42, 45, 66, 150, 184, 186, 188, 222] [19, 40, 43, 46, 67, 151, 185, 187, 189, 223] [20, 41, 44, 47, 68, 152, 186, 188, 190, 224] [21, 36, 42, 45, 69, 153, 187, 189, 191, 225] [22, 37, 43, 46, 70, 154, 180, 188, 190, 226] [23, 38, 44, 47, 71, 155, 181, 189, 191, 227] [24, 48, 51, 57, 72, 156, 192, 194, 202, 228] [25, 49, 52, 58, 73, 157, 193, 195, 203, 229] [26, 50, 53, 59, 74, 158, 192, 194, 196, 230] [27, 48, 51, 54, 75, 159, 193, 195, 197, 231] [28, 49, 52, 55, 76, 160, 194, 196, 198, 232] [29, 50, 53, 56, 77, 161, 195, 197, 199, 233] [30, 51, 54, 57, 78, 162, 196, 198, 200, 234] [31, 52, 55, 58, 79, 163, 197, 199, 201, 235] [32, 53, 56, 59, 80, 164, 198, 200, 202, 236] [33, 48, 54, 57, 81, 165, 199, 201, 203, 237] [34, 49, 55, 58, 82, 166, 192, 200, 202, 238] [35, 50, 56, 59, 83, 167, 193, 201, 203, 239] [36, 60, 63, 69, 84, 168, 204, 206, 214, 240] [37, 61, 64, 70, 85, 169, 205, 207, 215, 241] [38, 62, 65, 71, 86, 170, 204, 206, 208, 242] [39, 60, 63, 66, 87, 171, 205, 207, 209, 243] [40, 61, 64, 67, 88, 172, 206, 208, 210, 244] [41, 62, 65, 68, 89, 173, 207, 209, 211, 245] [42, 63, 66, 69, 90, 174, 208, 210, 212, 246] [43, 64, 67, 70, 91, 175, 209, 211, 213, 247] [44, 65, 68, 71, 92, 176, 210, 212, 214, 248] [45, 60, 66, 69, 93, 177, 211, 213, 215, 249] [46, 61, 67, 70, 94, 178, 204, 212, 214, 250] [47, 62, 68, 71, 95, 179, 205, 213, 215, 251] [48, 72, 75, 81, 96, 180, 216, 218, 226, 252] [49, 73, 76, 82, 97, 181, 217, 219, 227, 253] [50, 74, 77, 83, 98, 182, 216, 218, 220, 254] [51, 72, 75, 78, 99, 183, 217, 219, 221, 255] [52, 73, 76, 79, 100, 184, 218, 220, 222, 256] [53, 74, 77, 80, 101, 185, 219, 221, 223, 257] [54, 75, 78, 81, 102, 186, 220, 222, 224, 258] [55, 76, 79, 82, 103, 187, 221, 223, 225, 259] [56, 77, 80, 83, 104, 188, 222, 224, 226, 260] [57, 72, 78, 81, 105, 189, 223, 225, 227, 261] [58, 73, 79, 82, 106, 190, 216, 224, 226, 262] [59, 74, 80, 83, 107, 191, 217, 225, 227, 263] [60, 84, 87, 93, 108, 192, 228, 230, 238, 264] [61, 85, 88, 94, 109, 193, 229, 231, 239, 265] [62, 86, 89, 95, 110, 194, 228, 230, 232, 266] [63, 84, 87, 90, 111, 195, 229, 231, 233, 267] [64, 85, 88, 91, 112, 196, 230, 232, 234, 268] [65, 86, 89, 92, 113, 197, 231, 233, 235, 269] [66, 87, 90, 93, 114, 198, 232, 234, 236, 270] [67, 88, 91, 94, 115, 199, 233, 235, 237, 271] [68, 89, 92, 95, 116, 200, 234, 236, 238, 272] [69, 84, 90, 93, 117, 201, 235, 237, 239, 273] [70, 85, 91, 94, 118, 202, 228, 236, 238, 274] [71, 86, 92, 95, 119, 203, 229, 237, 239, 275] [72, 96, 99, 105, 120, 204, 240, 242, 250, 276] [73, 97, 100, 106, 121, 205, 241, 243, 251, 277] [74, 98, 101, 107, 122, 206, 240, 242, 244, 278] [75, 96, 99, 102, 123, 207, 241, 243, 245, 279] [76, 97, 100, 103, 124, 208, 242, 244, 246, 280] [77, 98, 101, 104, 125, 209, 243, 245, 247, 281] [78, 99, 102, 105, 126, 210, 244, 246, 248, 282] [79, 100, 103, 106, 127, 211, 245, 247, 249, 283] [80, 101, 104, 107, 128, 212, 246, 248, 250, 284] [81, 96, 102, 105, 129, 213, 247, 249, 251, 285] [82, 97, 103, 106, 130, 214, 240, 248, 250, 286] [83, 98, 104, 107, 131, 215, 241, 249, 251, 287] [84, 108, 111, 117, 132, 144, 216, 252, 254, 262] [85, 109, 112, 118, 133, 145, 217, 253, 255, 263] [86, 110, 113, 119, 134, 146, 218, 252, 254, 256] [87, 108, 111, 114, 135, 147, 219, 253, 255, 257] [88, 109, 112, 115, 136, 148, 220, 254, 256, 258] [89, 110, 113, 116, 137, 149, 221, 255, 257, 259] [90, 111, 114, 117, 138, 150, 222, 256, 258, 260] [91, 112, 115, 118, 139, 151, 223, 257, 259, 261] [92, 113, 116, 119, 140, 152, 224, 258, 260, 262] [93, 108, 114, 117, 141, 153, 225, 259, 261, 263] [94, 109, 115, 118, 142, 154, 226, 252, 260, 262] [95, 110, 116, 119, 143, 155, 227, 253, 261, 263] [0, 96, 120, 123, 129, 156, 228, 264, 266, 274] [1, 97, 121, 124, 130, 157, 229, 265, 267, 275] [2, 98, 122, 125, 131, 158, 230, 264, 266, 268] [3, 99, 120, 123, 126, 159, 231, 265, 267, 269] [4, 100, 121, 124, 127, 160, 232, 266, 268, 270] [5, 101, 122, 125, 128, 161, 233, 267, 269, 271] [6, 102, 123, 126, 129, 162, 234, 268, 270, 272] [7, 103, 124, 127, 130, 163, 235, 269, 271, 273] [8, 104, 125, 128, 131, 164, 236, 270, 272, 274] [9, 105, 120, 126, 129, 165, 237, 271, 273, 275] [10, 106, 121, 127, 130, 166, 238, 264, 272, 274] [11, 107, 122, 128, 131, 167, 239, 265, 273, 275] [12, 108, 132, 135, 141, 168, 240, 276, 278, 286] [13, 109, 133, 136, 142, 169, 241, 277, 279, 287] [14, 110, 134, 137, 143, 170, 242, 276, 278, 280] [15, 111, 132, 135, 138, 171, 243, 277, 279, 281] [16, 112, 133, 136, 139, 172, 244, 278, 280, 282] [17, 113, 134, 137, 140, 173, 245, 279, 281, 283] [18, 114, 135, 138, 141, 174, 246, 280, 282, 284] [19, 115, 136, 139, 142, 175, 247, 281, 283, 285] [20, 116, 137, 140, 143, 176, 248, 282, 284, 286] [21, 117, 132, 138, 141, 177, 249, 283, 285, 287] [22, 118, 133, 139, 142, 178, 250, 276, 284, 286] [23, 119, 134, 140, 143, 179, 251, 277, 285, 287]
H_Z (144 checks, sparse supports)
[0, 2, 10, 36, 108, 144, 147, 153, 168, 264] [1, 3, 11, 37, 109, 145, 148, 154, 169, 265] [0, 2, 4, 38, 110, 146, 149, 155, 170, 266] [1, 3, 5, 39, 111, 144, 147, 150, 171, 267] [2, 4, 6, 40, 112, 145, 148, 151, 172, 268] [3, 5, 7, 41, 113, 146, 149, 152, 173, 269] [4, 6, 8, 42, 114, 147, 150, 153, 174, 270] [5, 7, 9, 43, 115, 148, 151, 154, 175, 271] [6, 8, 10, 44, 116, 149, 152, 155, 176, 272] [7, 9, 11, 45, 117, 144, 150, 153, 177, 273] [0, 8, 10, 46, 118, 145, 151, 154, 178, 274] [1, 9, 11, 47, 119, 146, 152, 155, 179, 275] [12, 14, 22, 48, 120, 156, 159, 165, 180, 276] [13, 15, 23, 49, 121, 157, 160, 166, 181, 277] [12, 14, 16, 50, 122, 158, 161, 167, 182, 278] [13, 15, 17, 51, 123, 156, 159, 162, 183, 279] [14, 16, 18, 52, 124, 157, 160, 163, 184, 280] [15, 17, 19, 53, 125, 158, 161, 164, 185, 281] [16, 18, 20, 54, 126, 159, 162, 165, 186, 282] [17, 19, 21, 55, 127, 160, 163, 166, 187, 283] [18, 20, 22, 56, 128, 161, 164, 167, 188, 284] [19, 21, 23, 57, 129, 156, 162, 165, 189, 285] [12, 20, 22, 58, 130, 157, 163, 166, 190, 286] [13, 21, 23, 59, 131, 158, 164, 167, 191, 287] [24, 26, 34, 60, 132, 144, 168, 171, 177, 192] [25, 27, 35, 61, 133, 145, 169, 172, 178, 193] [24, 26, 28, 62, 134, 146, 170, 173, 179, 194] [25, 27, 29, 63, 135, 147, 168, 171, 174, 195] [26, 28, 30, 64, 136, 148, 169, 172, 175, 196] [27, 29, 31, 65, 137, 149, 170, 173, 176, 197] [28, 30, 32, 66, 138, 150, 171, 174, 177, 198] [29, 31, 33, 67, 139, 151, 172, 175, 178, 199] [30, 32, 34, 68, 140, 152, 173, 176, 179, 200] [31, 33, 35, 69, 141, 153, 168, 174, 177, 201] [24, 32, 34, 70, 142, 154, 169, 175, 178, 202] [25, 33, 35, 71, 143, 155, 170, 176, 179, 203] [0, 36, 38, 46, 72, 156, 180, 183, 189, 204] [1, 37, 39, 47, 73, 157, 181, 184, 190, 205] [2, 36, 38, 40, 74, 158, 182, 185, 191, 206] [3, 37, 39, 41, 75, 159, 180, 183, 186, 207] [4, 38, 40, 42, 76, 160, 181, 184, 187, 208] [5, 39, 41, 43, 77, 161, 182, 185, 188, 209] [6, 40, 42, 44, 78, 162, 183, 186, 189, 210] [7, 41, 43, 45, 79, 163, 184, 187, 190, 211] [8, 42, 44, 46, 80, 164, 185, 188, 191, 212] [9, 43, 45, 47, 81, 165, 180, 186, 189, 213] [10, 36, 44, 46, 82, 166, 181, 187, 190, 214] [11, 37, 45, 47, 83, 167, 182, 188, 191, 215] [12, 48, 50, 58, 84, 168, 192, 195, 201, 216] [13, 49, 51, 59, 85, 169, 193, 196, 202, 217] [14, 48, 50, 52, 86, 170, 194, 197, 203, 218] [15, 49, 51, 53, 87, 171, 192, 195, 198, 219] [16, 50, 52, 54, 88, 172, 193, 196, 199, 220] [17, 51, 53, 55, 89, 173, 194, 197, 200, 221] [18, 52, 54, 56, 90, 174, 195, 198, 201, 222] [19, 53, 55, 57, 91, 175, 196, 199, 202, 223] [20, 54, 56, 58, 92, 176, 197, 200, 203, 224] [21, 55, 57, 59, 93, 177, 192, 198, 201, 225] [22, 48, 56, 58, 94, 178, 193, 199, 202, 226] [23, 49, 57, 59, 95, 179, 194, 200, 203, 227] [24, 60, 62, 70, 96, 180, 204, 207, 213, 228] [25, 61, 63, 71, 97, 181, 205, 208, 214, 229] [26, 60, 62, 64, 98, 182, 206, 209, 215, 230] [27, 61, 63, 65, 99, 183, 204, 207, 210, 231] [28, 62, 64, 66, 100, 184, 205, 208, 211, 232] [29, 63, 65, 67, 101, 185, 206, 209, 212, 233] [30, 64, 66, 68, 102, 186, 207, 210, 213, 234] [31, 65, 67, 69, 103, 187, 208, 211, 214, 235] [32, 66, 68, 70, 104, 188, 209, 212, 215, 236] [33, 67, 69, 71, 105, 189, 204, 210, 213, 237] [34, 60, 68, 70, 106, 190, 205, 211, 214, 238] [35, 61, 69, 71, 107, 191, 206, 212, 215, 239] [36, 72, 74, 82, 108, 192, 216, 219, 225, 240] [37, 73, 75, 83, 109, 193, 217, 220, 226, 241] [38, 72, 74, 76, 110, 194, 218, 221, 227, 242] [39, 73, 75, 77, 111, 195, 216, 219, 222, 243] [40, 74, 76, 78, 112, 196, 217, 220, 223, 244] [41, 75, 77, 79, 113, 197, 218, 221, 224, 245] [42, 76, 78, 80, 114, 198, 219, 222, 225, 246] [43, 77, 79, 81, 115, 199, 220, 223, 226, 247] [44, 78, 80, 82, 116, 200, 221, 224, 227, 248] [45, 79, 81, 83, 117, 201, 216, 222, 225, 249] [46, 72, 80, 82, 118, 202, 217, 223, 226, 250] [47, 73, 81, 83, 119, 203, 218, 224, 227, 251] [48, 84, 86, 94, 120, 204, 228, 231, 237, 252] [49, 85, 87, 95, 121, 205, 229, 232, 238, 253] [50, 84, 86, 88, 122, 206, 230, 233, 239, 254] [51, 85, 87, 89, 123, 207, 228, 231, 234, 255] [52, 86, 88, 90, 124, 208, 229, 232, 235, 256] [53, 87, 89, 91, 125, 209, 230, 233, 236, 257] [54, 88, 90, 92, 126, 210, 231, 234, 237, 258] [55, 89, 91, 93, 127, 211, 232, 235, 238, 259] [56, 90, 92, 94, 128, 212, 233, 236, 239, 260] [57, 91, 93, 95, 129, 213, 228, 234, 237, 261] [58, 84, 92, 94, 130, 214, 229, 235, 238, 262] [59, 85, 93, 95, 131, 215, 230, 236, 239, 263] [60, 96, 98, 106, 132, 216, 240, 243, 249, 264] [61, 97, 99, 107, 133, 217, 241, 244, 250, 265] [62, 96, 98, 100, 134, 218, 242, 245, 251, 266] [63, 97, 99, 101, 135, 219, 240, 243, 246, 267] [64, 98, 100, 102, 136, 220, 241, 244, 247, 268] [65, 99, 101, 103, 137, 221, 242, 245, 248, 269] [66, 100, 102, 104, 138, 222, 243, 246, 249, 270] [67, 101, 103, 105, 139, 223, 244, 247, 250, 271] [68, 102, 104, 106, 140, 224, 245, 248, 251, 272] [69, 103, 105, 107, 141, 225, 240, 246, 249, 273] [70, 96, 104, 106, 142, 226, 241, 247, 250, 274] [71, 97, 105, 107, 143, 227, 242, 248, 251, 275] [0, 72, 108, 110, 118, 228, 252, 255, 261, 276] [1, 73, 109, 111, 119, 229, 253, 256, 262, 277] [2, 74, 108, 110, 112, 230, 254, 257, 263, 278] [3, 75, 109, 111, 113, 231, 252, 255, 258, 279] [4, 76, 110, 112, 114, 232, 253, 256, 259, 280] [5, 77, 111, 113, 115, 233, 254, 257, 260, 281] [6, 78, 112, 114, 116, 234, 255, 258, 261, 282] [7, 79, 113, 115, 117, 235, 256, 259, 262, 283] [8, 80, 114, 116, 118, 236, 257, 260, 263, 284] [9, 81, 115, 117, 119, 237, 252, 258, 261, 285] [10, 82, 108, 116, 118, 238, 253, 259, 262, 286] [11, 83, 109, 117, 119, 239, 254, 260, 263, 287] [12, 84, 120, 122, 130, 144, 240, 264, 267, 273] [13, 85, 121, 123, 131, 145, 241, 265, 268, 274] [14, 86, 120, 122, 124, 146, 242, 266, 269, 275] [15, 87, 121, 123, 125, 147, 243, 264, 267, 270] [16, 88, 122, 124, 126, 148, 244, 265, 268, 271] [17, 89, 123, 125, 127, 149, 245, 266, 269, 272] [18, 90, 124, 126, 128, 150, 246, 267, 270, 273] [19, 91, 125, 127, 129, 151, 247, 268, 271, 274] [20, 92, 126, 128, 130, 152, 248, 269, 272, 275] [21, 93, 127, 129, 131, 153, 249, 264, 270, 273] [22, 94, 120, 128, 130, 154, 250, 265, 271, 274] [23, 95, 121, 129, 131, 155, 251, 266, 272, 275] [24, 96, 132, 134, 142, 156, 252, 276, 279, 285] [25, 97, 133, 135, 143, 157, 253, 277, 280, 286] [26, 98, 132, 134, 136, 158, 254, 278, 281, 287] [27, 99, 133, 135, 137, 159, 255, 276, 279, 282] [28, 100, 134, 136, 138, 160, 256, 277, 280, 283] [29, 101, 135, 137, 139, 161, 257, 278, 281, 284] [30, 102, 136, 138, 140, 162, 258, 279, 282, 285] [31, 103, 137, 139, 141, 163, 259, 280, 283, 286] [32, 104, 138, 140, 142, 164, 260, 281, 284, 287] [33, 105, 139, 141, 143, 165, 261, 276, 282, 285] [34, 106, 132, 140, 142, 166, 262, 277, 283, 286] [35, 107, 133, 141, 143, 167, 263, 278, 284, 287]
raw submission JSON