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[[330,8,24]] d ≤
n
330
k
8
d
24
kd²/n
13.964
w
6

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Distance

d_X 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[8, 34, 36, 65, 86, 96, 100, 110, 128, 149, 156, 167, 174, 210, 260, 271, 274, 280, 281, 285, 289, 294, 314, 326]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[11, 32, 69, 72, 78, 87, 96, 112, 122, 131, 137, 142, 146, 159, 176, 191, 210, 220, 225, 288, 293, 296, 298, 305]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors Zijian Liang and Ke Liu and Hao Song and Yu-An Chen
provenance literature baseline
construction Twisted-torus bivariate-bicycle code from Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025). Stabilizers f(x,y)=1+x+x-6y2, g(x,y)=1+y+x2y5 on the abelian quotient group Z2/L with twist basis a_1=[0, 55], a_2=[3, 23] (n=2|det[a_1,a_2]|=2*165). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].
model classical construction (no AI model)
reference arXiv:2503.03827
date 2025
notes Reconstructed baseline from Liang, Liu, Song, Chen, Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025), arXiv:2503.03827. The [[n,k,d]] parameter set and stabilizer polynomials are published in that paper; this entry reproduces them. Distance is an upper bound per the paper (probabilistic for d>20); the witness is the surrogate logical of that weight. Seeded to fill a board coverage gap at this block size, not a discovery.
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (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 165 · Z-checks 165
H_X (165 checks, sparse supports)
[0, 1, 115, 165, 167, 198] [1, 3, 124, 166, 169, 206] [2, 4, 125, 167, 170, 207] [3, 6, 133, 168, 172, 215] [4, 7, 134, 169, 173, 216] [5, 8, 99, 170, 174, 217] [6, 10, 142, 171, 176, 224] [7, 11, 143, 172, 177, 225] [8, 12, 108, 173, 178, 226] [9, 13, 109, 174, 179, 227] [10, 15, 150, 175, 181, 233] [11, 16, 151, 176, 182, 234] [12, 17, 117, 177, 183, 235] [13, 18, 118, 178, 184, 236] [14, 19, 119, 179, 185, 201] [15, 21, 156, 180, 187, 242] [16, 22, 157, 181, 188, 243] [17, 23, 126, 182, 189, 244] [18, 24, 127, 183, 190, 245] [19, 25, 128, 184, 191, 210] [20, 26, 129, 185, 192, 211] [5, 21, 28, 186, 194, 251] [22, 29, 161, 187, 195, 252] [23, 30, 135, 188, 196, 253] [24, 31, 136, 189, 197, 254] [25, 32, 137, 190, 198, 219] [26, 33, 138, 191, 199, 220] [27, 34, 139, 192, 200, 221] [8, 28, 36, 193, 202, 260] [9, 29, 37, 194, 203, 261] [30, 38, 144, 195, 204, 262] [31, 39, 145, 196, 205, 263] [32, 40, 146, 197, 206, 228] [33, 41, 147, 198, 207, 229] [34, 42, 148, 199, 208, 230] [0, 35, 43, 200, 209, 231] [12, 36, 45, 201, 211, 269] [13, 37, 46, 202, 212, 270] [14, 38, 47, 203, 213, 271] [39, 48, 152, 204, 214, 272] [40, 49, 153, 205, 215, 237] [41, 50, 154, 206, 216, 238] [42, 51, 155, 207, 217, 239] [1, 43, 52, 208, 218, 240] [2, 44, 53, 186, 209, 241] [17, 45, 54, 210, 220, 278] [18, 46, 55, 211, 221, 279] [19, 47, 56, 212, 222, 280] [20, 48, 57, 213, 223, 281] [49, 58, 158, 214, 224, 246] [50, 59, 159, 215, 225, 247] [51, 60, 160, 216, 226, 248] [3, 52, 61, 217, 227, 249] [4, 53, 62, 193, 218, 250] [23, 54, 63, 219, 229, 287] [24, 55, 64, 220, 230, 288] [25, 56, 65, 221, 231, 289] [26, 57, 66, 222, 232, 290] [27, 58, 67, 223, 233, 255] [59, 68, 162, 224, 234, 256] [60, 69, 163, 225, 235, 257] [6, 61, 70, 226, 236, 258] [7, 62, 71, 201, 227, 259] [30, 63, 72, 228, 238, 296] [31, 64, 73, 229, 239, 297] [32, 65, 74, 230, 240, 298] [33, 66, 75, 231, 241, 299] [34, 67, 76, 232, 242, 264] [35, 68, 77, 233, 243, 265] [69, 78, 164, 234, 244, 266] [10, 70, 79, 235, 245, 267] [11, 71, 80, 210, 236, 268] [38, 72, 81, 237, 247, 305] [39, 73, 82, 238, 248, 306] [40, 74, 83, 239, 249, 307] [41, 75, 84, 240, 250, 308] [42, 76, 85, 241, 251, 273] [43, 77, 86, 242, 252, 274] [44, 78, 87, 243, 253, 275] [15, 79, 88, 244, 254, 276] [16, 80, 89, 219, 245, 277] [47, 81, 90, 165, 246, 256] [48, 82, 91, 247, 257, 314] [49, 83, 92, 248, 258, 315] [50, 84, 93, 249, 259, 316] [51, 85, 94, 250, 260, 282] [52, 86, 95, 251, 261, 283] [53, 87, 96, 252, 262, 284] [21, 88, 97, 253, 263, 285] [22, 89, 98, 228, 254, 286] [56, 90, 99, 166, 255, 265] [57, 91, 100, 167, 256, 266] [58, 92, 101, 257, 267, 321] [59, 93, 102, 258, 268, 322] [60, 94, 103, 259, 269, 291] [61, 95, 104, 260, 270, 292] [62, 96, 105, 261, 271, 293] [28, 97, 106, 262, 272, 294] [29, 98, 107, 237, 263, 295] [65, 99, 108, 168, 264, 274] [66, 100, 109, 169, 265, 275] [67, 101, 110, 170, 266, 276] [68, 102, 111, 267, 277, 326] [69, 103, 112, 268, 278, 300] [70, 104, 113, 269, 279, 301] [71, 105, 114, 270, 280, 302] [36, 106, 115, 271, 281, 303] [37, 107, 116, 246, 272, 304] [74, 108, 117, 171, 273, 283] [75, 109, 118, 172, 274, 284] [76, 110, 119, 173, 275, 285] [77, 111, 120, 174, 276, 286] [78, 112, 121, 277, 287, 309] [79, 113, 122, 278, 288, 310] [80, 114, 123, 279, 289, 311] [45, 115, 124, 280, 290, 312] [46, 116, 125, 255, 281, 313] [83, 117, 126, 175, 282, 292] [84, 118, 127, 176, 283, 293] [85, 119, 128, 177, 284, 294] [86, 120, 129, 178, 285, 295] [87, 121, 130, 179, 286, 296] [88, 122, 131, 287, 297, 317] [89, 123, 132, 288, 298, 318] [54, 124, 133, 289, 299, 319] [55, 125, 134, 264, 290, 320] [92, 126, 135, 180, 291, 301] [93, 127, 136, 181, 292, 302] [94, 128, 137, 182, 293, 303] [95, 129, 138, 183, 294, 304] [96, 130, 139, 184, 295, 305] [97, 131, 140, 185, 296, 306] [98, 132, 141, 297, 307, 323] [63, 133, 142, 298, 308, 324] [64, 134, 143, 273, 299, 325] [101, 135, 144, 186, 300, 310] [102, 136, 145, 187, 301, 311] [103, 137, 146, 188, 302, 312] [104, 138, 147, 189, 303, 313] [105, 139, 148, 165, 190, 304] [0, 106, 140, 191, 305, 314] [107, 141, 149, 192, 306, 315] [72, 142, 150, 307, 316, 327] [73, 143, 151, 282, 308, 328] [14, 110, 144, 193, 309, 317] [111, 145, 152, 194, 310, 318] [112, 146, 153, 195, 311, 319] [113, 147, 154, 196, 312, 320] [114, 148, 155, 166, 197, 313] [2, 116, 149, 199, 314, 321] [81, 150, 156, 200, 315, 322] [82, 151, 157, 291, 316, 329] [20, 120, 152, 202, 317, 323] [121, 153, 158, 203, 318, 324] [122, 154, 159, 204, 319, 325] [123, 155, 160, 168, 205, 320] [5, 90, 156, 208, 321, 326] [91, 157, 161, 209, 300, 322] [27, 130, 158, 212, 323, 327] [131, 159, 162, 213, 324, 328] [132, 160, 163, 171, 214, 325] [9, 100, 161, 218, 309, 326] [35, 140, 162, 222, 327, 329] [141, 163, 164, 175, 223, 328] [44, 149, 164, 180, 232, 329]
H_Z (165 checks, sparse supports)
[0, 81, 139, 165, 200, 305] [1, 90, 148, 165, 166, 208] [0, 2, 91, 167, 209, 314] [3, 99, 155, 166, 168, 217] [1, 4, 100, 167, 169, 218] [2, 5, 101, 170, 186, 321] [6, 108, 160, 168, 171, 226] [3, 7, 109, 169, 172, 227] [4, 8, 110, 170, 173, 193] [5, 9, 111, 174, 194, 326] [10, 117, 163, 171, 175, 235] [6, 11, 118, 172, 176, 236] [7, 12, 119, 173, 177, 201] [8, 13, 120, 174, 178, 202] [9, 14, 121, 179, 203, 309] [15, 126, 164, 175, 180, 244] [10, 16, 127, 176, 181, 245] [11, 17, 128, 177, 182, 210] [12, 18, 129, 178, 183, 211] [13, 19, 130, 179, 184, 212] [14, 20, 131, 185, 213, 317] [21, 44, 135, 180, 186, 253] [15, 22, 136, 181, 187, 254] [16, 23, 137, 182, 188, 219] [17, 24, 138, 183, 189, 220] [18, 25, 139, 184, 190, 221] [19, 26, 140, 185, 191, 222] [20, 27, 141, 192, 223, 323] [28, 53, 144, 186, 193, 262] [21, 29, 145, 187, 194, 263] [22, 30, 146, 188, 195, 228] [23, 31, 147, 189, 196, 229] [24, 32, 148, 190, 197, 230] [0, 25, 33, 191, 198, 231] [26, 34, 149, 192, 199, 232] [27, 35, 150, 200, 233, 327] [14, 36, 62, 193, 201, 271] [28, 37, 152, 194, 202, 272] [29, 38, 153, 195, 203, 237] [30, 39, 154, 196, 204, 238] [31, 40, 155, 197, 205, 239] [1, 32, 41, 198, 206, 240] [2, 33, 42, 199, 207, 241] [34, 43, 156, 200, 208, 242] [35, 44, 157, 209, 243, 329] [19, 45, 71, 201, 210, 280] [20, 36, 46, 202, 211, 281] [37, 47, 158, 203, 212, 246] [38, 48, 159, 204, 213, 247] [39, 49, 160, 205, 214, 248] [3, 40, 50, 206, 215, 249] [4, 41, 51, 207, 216, 250] [5, 42, 52, 208, 217, 251] [43, 53, 161, 209, 218, 252] [25, 54, 80, 210, 219, 289] [26, 45, 55, 211, 220, 290] [27, 46, 56, 212, 221, 255] [47, 57, 162, 213, 222, 256] [48, 58, 163, 214, 223, 257] [6, 49, 59, 215, 224, 258] [7, 50, 60, 216, 225, 259] [8, 51, 61, 217, 226, 260] [9, 52, 62, 218, 227, 261] [32, 63, 89, 219, 228, 298] [33, 54, 64, 220, 229, 299] [34, 55, 65, 221, 230, 264] [35, 56, 66, 222, 231, 265] [57, 67, 164, 223, 232, 266] [10, 58, 68, 224, 233, 267] [11, 59, 69, 225, 234, 268] [12, 60, 70, 226, 235, 269] [13, 61, 71, 227, 236, 270] [40, 72, 98, 228, 237, 307] [41, 63, 73, 229, 238, 308] [42, 64, 74, 230, 239, 273] [43, 65, 75, 231, 240, 274] [44, 66, 76, 232, 241, 275] [15, 67, 77, 233, 242, 276] [16, 68, 78, 234, 243, 277] [17, 69, 79, 235, 244, 278] [18, 70, 80, 236, 245, 279] [49, 81, 107, 237, 246, 315] [50, 72, 82, 238, 247, 316] [51, 73, 83, 239, 248, 282] [52, 74, 84, 240, 249, 283] [53, 75, 85, 241, 250, 284] [21, 76, 86, 242, 251, 285] [22, 77, 87, 243, 252, 286] [23, 78, 88, 244, 253, 287] [24, 79, 89, 245, 254, 288] [58, 90, 116, 246, 255, 321] [59, 81, 91, 247, 256, 322] [60, 82, 92, 248, 257, 291] [61, 83, 93, 249, 258, 292] [62, 84, 94, 250, 259, 293] [28, 85, 95, 251, 260, 294] [29, 86, 96, 252, 261, 295] [30, 87, 97, 253, 262, 296] [31, 88, 98, 254, 263, 297] [67, 99, 125, 170, 255, 264] [68, 90, 100, 256, 265, 326] [69, 91, 101, 257, 266, 300] [70, 92, 102, 258, 267, 301] [71, 93, 103, 259, 268, 302] [36, 94, 104, 260, 269, 303] [37, 95, 105, 261, 270, 304] [38, 96, 106, 262, 271, 305] [39, 97, 107, 263, 272, 306] [76, 108, 134, 173, 264, 273] [77, 99, 109, 174, 265, 274] [78, 100, 110, 266, 275, 309] [79, 101, 111, 267, 276, 310] [80, 102, 112, 268, 277, 311] [45, 103, 113, 269, 278, 312] [46, 104, 114, 270, 279, 313] [47, 105, 115, 165, 271, 280] [48, 106, 116, 272, 281, 314] [85, 117, 143, 177, 273, 282] [86, 108, 118, 178, 274, 283] [87, 109, 119, 179, 275, 284] [88, 110, 120, 276, 285, 317] [89, 111, 121, 277, 286, 318] [54, 112, 122, 278, 287, 319] [55, 113, 123, 279, 288, 320] [56, 114, 124, 166, 280, 289] [57, 115, 125, 167, 281, 290] [94, 126, 151, 182, 282, 291] [95, 117, 127, 183, 283, 292] [96, 118, 128, 184, 284, 293] [97, 119, 129, 185, 285, 294] [98, 120, 130, 286, 295, 323] [63, 121, 131, 287, 296, 324] [64, 122, 132, 288, 297, 325] [65, 123, 133, 168, 289, 298] [66, 124, 134, 169, 290, 299] [103, 135, 157, 188, 291, 300] [104, 126, 136, 189, 292, 301] [105, 127, 137, 190, 293, 302] [106, 128, 138, 191, 294, 303] [107, 129, 139, 192, 295, 304] [72, 130, 140, 296, 305, 327] [73, 131, 141, 297, 306, 328] [74, 132, 142, 171, 298, 307] [75, 133, 143, 172, 299, 308] [112, 144, 161, 195, 300, 309] [113, 135, 145, 196, 301, 310] [114, 136, 146, 197, 302, 311] [115, 137, 147, 198, 303, 312] [116, 138, 148, 199, 304, 313] [82, 140, 149, 306, 314, 329] [83, 141, 150, 175, 307, 315] [84, 142, 151, 176, 308, 316] [122, 144, 152, 204, 310, 317] [123, 145, 153, 205, 311, 318] [124, 146, 154, 206, 312, 319] [125, 147, 155, 207, 313, 320] [92, 149, 156, 180, 315, 321] [93, 150, 157, 181, 316, 322] [132, 152, 158, 214, 318, 323] [133, 153, 159, 215, 319, 324] [134, 154, 160, 216, 320, 325] [102, 156, 161, 187, 322, 326] [142, 158, 162, 224, 324, 327] [143, 159, 163, 225, 325, 328] [151, 162, 164, 234, 328, 329]
raw submission JSON