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[[340,16,18]] d ≤
n
340
k
16
d
18
kd²/n
15.247
w
6

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Distance

d_X 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[25, 33, 35, 53, 59, 74, 79, 81, 206, 222, 233, 237, 238, 249, 256, 263, 270, 272]
d_Z 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[22, 24, 27, 32, 38, 39, 49, 54, 166, 168, 169, 197, 213, 217, 220, 223, 243, 251]
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+y-4, g(x,y)=1+y+x4 on the abelian quotient group Z2/L with twist basis a_1=[0, 34], a_2=[5, -7] (n=2|det[a_1,a_2]|=2*170). 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 170 · Z-checks 170
H_X (170 checks, sparse supports)
[0, 1, 135, 170, 172, 180] [1, 3, 142, 171, 174, 185] [2, 4, 143, 172, 175, 186] [3, 6, 149, 173, 177, 191] [4, 7, 150, 174, 178, 192] [5, 8, 151, 175, 179, 193] [6, 10, 155, 176, 181, 198] [7, 11, 156, 177, 182, 199] [8, 12, 157, 178, 183, 200] [9, 13, 158, 179, 184, 201] [10, 15, 160, 180, 186, 205] [11, 16, 161, 181, 187, 206] [12, 17, 162, 182, 188, 207] [13, 18, 163, 183, 189, 208] [0, 14, 19, 184, 190, 209] [9, 15, 21, 185, 192, 212] [16, 22, 164, 186, 193, 213] [17, 23, 165, 187, 194, 214] [18, 24, 166, 188, 195, 215] [1, 19, 25, 189, 196, 216] [2, 20, 26, 190, 197, 217] [13, 21, 28, 191, 199, 219] [14, 22, 29, 192, 200, 220] [23, 30, 167, 193, 201, 221] [24, 31, 168, 194, 202, 222] [3, 25, 32, 195, 203, 223] [4, 26, 33, 196, 204, 224] [5, 27, 34, 185, 197, 225] [18, 28, 35, 198, 206, 226] [19, 29, 36, 199, 207, 227] [20, 30, 37, 200, 208, 228] [31, 38, 169, 201, 209, 229] [6, 32, 39, 202, 210, 230] [7, 33, 40, 203, 211, 231] [8, 34, 41, 191, 204, 232] [24, 35, 42, 205, 213, 233] [25, 36, 43, 206, 214, 234] [26, 37, 44, 207, 215, 235] [27, 38, 45, 208, 216, 236] [10, 39, 46, 209, 217, 237] [11, 40, 47, 210, 218, 238] [12, 41, 48, 198, 211, 239] [31, 42, 49, 212, 220, 240] [32, 43, 50, 213, 221, 241] [33, 44, 51, 214, 222, 242] [34, 45, 52, 215, 223, 243] [15, 46, 53, 216, 224, 244] [16, 47, 54, 217, 225, 245] [17, 48, 55, 205, 218, 246] [38, 49, 56, 219, 227, 247] [39, 50, 57, 220, 228, 248] [40, 51, 58, 221, 229, 249] [41, 52, 59, 222, 230, 250] [21, 53, 60, 223, 231, 251] [22, 54, 61, 224, 232, 252] [23, 55, 62, 212, 225, 253] [45, 56, 63, 226, 234, 254] [46, 57, 64, 227, 235, 255] [47, 58, 65, 228, 236, 256] [48, 59, 66, 229, 237, 257] [28, 60, 67, 230, 238, 258] [29, 61, 68, 231, 239, 259] [30, 62, 69, 219, 232, 260] [52, 63, 70, 233, 241, 261] [53, 64, 71, 234, 242, 262] [54, 65, 72, 235, 243, 263] [55, 66, 73, 236, 244, 264] [35, 67, 74, 237, 245, 265] [36, 68, 75, 238, 246, 266] [37, 69, 76, 226, 239, 267] [59, 70, 77, 240, 248, 268] [60, 71, 78, 241, 249, 269] [61, 72, 79, 242, 250, 270] [62, 73, 80, 243, 251, 271] [42, 74, 81, 244, 252, 272] [43, 75, 82, 245, 253, 273] [44, 76, 83, 233, 246, 274] [66, 77, 84, 247, 255, 275] [67, 78, 85, 248, 256, 276] [68, 79, 86, 249, 257, 277] [69, 80, 87, 250, 258, 278] [49, 81, 88, 251, 259, 279] [50, 82, 89, 252, 260, 280] [51, 83, 90, 240, 253, 281] [73, 84, 91, 254, 262, 282] [74, 85, 92, 255, 263, 283] [75, 86, 93, 256, 264, 284] [76, 87, 94, 257, 265, 285] [56, 88, 95, 258, 266, 286] [57, 89, 96, 259, 267, 287] [58, 90, 97, 247, 260, 288] [80, 91, 98, 261, 269, 289] [81, 92, 99, 262, 270, 290] [82, 93, 100, 263, 271, 291] [83, 94, 101, 264, 272, 292] [63, 95, 102, 265, 273, 293] [64, 96, 103, 266, 274, 294] [65, 97, 104, 254, 267, 295] [87, 98, 105, 268, 276, 296] [88, 99, 106, 269, 277, 297] [89, 100, 107, 270, 278, 298] [90, 101, 108, 271, 279, 299] [70, 102, 109, 272, 280, 300] [71, 103, 110, 273, 281, 301] [72, 104, 111, 261, 274, 302] [94, 105, 112, 275, 283, 303] [95, 106, 113, 276, 284, 304] [96, 107, 114, 277, 285, 305] [97, 108, 115, 278, 286, 306] [77, 109, 116, 279, 287, 307] [78, 110, 117, 280, 288, 308] [79, 111, 118, 268, 281, 309] [101, 112, 119, 282, 290, 310] [102, 113, 120, 283, 291, 311] [103, 114, 121, 284, 292, 312] [104, 115, 122, 285, 293, 313] [84, 116, 123, 286, 294, 314] [85, 117, 124, 287, 295, 315] [86, 118, 125, 275, 288, 316] [108, 119, 126, 289, 297, 317] [109, 120, 127, 290, 298, 318] [110, 121, 128, 291, 299, 319] [111, 122, 129, 292, 300, 320] [91, 123, 130, 293, 301, 321] [92, 124, 131, 294, 302, 322] [93, 125, 132, 282, 295, 323] [115, 126, 133, 172, 296, 304] [116, 127, 134, 297, 305, 324] [117, 128, 135, 298, 306, 325] [118, 129, 136, 299, 307, 326] [98, 130, 137, 300, 308, 327] [99, 131, 138, 301, 309, 328] [100, 132, 139, 289, 302, 329] [122, 133, 140, 174, 303, 311] [123, 134, 141, 175, 304, 312] [124, 135, 142, 305, 313, 330] [125, 136, 143, 306, 314, 331] [105, 137, 144, 307, 315, 332] [106, 138, 145, 308, 316, 333] [107, 139, 146, 170, 296, 309] [129, 140, 147, 177, 310, 318] [130, 141, 148, 178, 311, 319] [131, 142, 149, 179, 312, 320] [132, 143, 150, 313, 321, 334] [112, 144, 151, 314, 322, 335] [113, 145, 152, 315, 323, 336] [114, 146, 153, 171, 303, 316] [2, 136, 147, 181, 317, 324] [137, 148, 154, 182, 318, 325] [138, 149, 155, 183, 319, 326] [139, 150, 156, 184, 320, 327] [119, 151, 157, 321, 328, 337] [120, 152, 158, 322, 329, 338] [121, 153, 159, 173, 310, 323] [5, 144, 154, 187, 324, 330] [145, 155, 160, 188, 325, 331] [146, 156, 161, 189, 326, 332] [126, 157, 162, 190, 327, 333] [127, 158, 163, 170, 328, 339] [0, 128, 159, 176, 317, 329] [9, 152, 160, 194, 330, 334] [153, 161, 164, 195, 331, 335] [133, 162, 165, 196, 332, 336] [134, 163, 166, 171, 197, 333] [14, 159, 164, 202, 334, 337] [140, 165, 167, 203, 335, 338] [141, 166, 168, 173, 204, 336] [20, 147, 167, 210, 337, 339] [148, 168, 169, 176, 211, 338] [27, 154, 169, 180, 218, 339]
H_Z (170 checks, sparse supports)
[0, 139, 158, 170, 184, 329] [1, 146, 163, 170, 171, 189] [0, 2, 126, 172, 190, 317] [3, 153, 166, 171, 173, 195] [1, 4, 133, 172, 174, 196] [2, 5, 134, 175, 197, 324] [6, 159, 168, 173, 176, 202] [3, 7, 140, 174, 177, 203] [4, 8, 141, 175, 178, 204] [5, 9, 142, 179, 185, 330] [0, 10, 169, 176, 180, 209] [6, 11, 147, 177, 181, 210] [7, 12, 148, 178, 182, 211] [8, 13, 149, 179, 183, 191] [9, 14, 150, 184, 192, 334] [1, 15, 27, 180, 185, 216] [2, 10, 16, 181, 186, 217] [11, 17, 154, 182, 187, 218] [12, 18, 155, 183, 188, 198] [13, 19, 156, 184, 189, 199] [14, 20, 157, 190, 200, 337] [3, 21, 34, 185, 191, 223] [4, 15, 22, 186, 192, 224] [5, 16, 23, 187, 193, 225] [17, 24, 160, 188, 194, 205] [18, 25, 161, 189, 195, 206] [19, 26, 162, 190, 196, 207] [20, 27, 163, 197, 208, 339] [6, 28, 41, 191, 198, 230] [7, 21, 29, 192, 199, 231] [8, 22, 30, 193, 200, 232] [9, 23, 31, 194, 201, 212] [24, 32, 164, 195, 202, 213] [25, 33, 165, 196, 203, 214] [26, 34, 166, 197, 204, 215] [10, 35, 48, 198, 205, 237] [11, 28, 36, 199, 206, 238] [12, 29, 37, 200, 207, 239] [13, 30, 38, 201, 208, 219] [14, 31, 39, 202, 209, 220] [32, 40, 167, 203, 210, 221] [33, 41, 168, 204, 211, 222] [15, 42, 55, 205, 212, 244] [16, 35, 43, 206, 213, 245] [17, 36, 44, 207, 214, 246] [18, 37, 45, 208, 215, 226] [19, 38, 46, 209, 216, 227] [20, 39, 47, 210, 217, 228] [40, 48, 169, 211, 218, 229] [21, 49, 62, 212, 219, 251] [22, 42, 50, 213, 220, 252] [23, 43, 51, 214, 221, 253] [24, 44, 52, 215, 222, 233] [25, 45, 53, 216, 223, 234] [26, 46, 54, 217, 224, 235] [27, 47, 55, 218, 225, 236] [28, 56, 69, 219, 226, 258] [29, 49, 57, 220, 227, 259] [30, 50, 58, 221, 228, 260] [31, 51, 59, 222, 229, 240] [32, 52, 60, 223, 230, 241] [33, 53, 61, 224, 231, 242] [34, 54, 62, 225, 232, 243] [35, 63, 76, 226, 233, 265] [36, 56, 64, 227, 234, 266] [37, 57, 65, 228, 235, 267] [38, 58, 66, 229, 236, 247] [39, 59, 67, 230, 237, 248] [40, 60, 68, 231, 238, 249] [41, 61, 69, 232, 239, 250] [42, 70, 83, 233, 240, 272] [43, 63, 71, 234, 241, 273] [44, 64, 72, 235, 242, 274] [45, 65, 73, 236, 243, 254] [46, 66, 74, 237, 244, 255] [47, 67, 75, 238, 245, 256] [48, 68, 76, 239, 246, 257] [49, 77, 90, 240, 247, 279] [50, 70, 78, 241, 248, 280] [51, 71, 79, 242, 249, 281] [52, 72, 80, 243, 250, 261] [53, 73, 81, 244, 251, 262] [54, 74, 82, 245, 252, 263] [55, 75, 83, 246, 253, 264] [56, 84, 97, 247, 254, 286] [57, 77, 85, 248, 255, 287] [58, 78, 86, 249, 256, 288] [59, 79, 87, 250, 257, 268] [60, 80, 88, 251, 258, 269] [61, 81, 89, 252, 259, 270] [62, 82, 90, 253, 260, 271] [63, 91, 104, 254, 261, 293] [64, 84, 92, 255, 262, 294] [65, 85, 93, 256, 263, 295] [66, 86, 94, 257, 264, 275] [67, 87, 95, 258, 265, 276] [68, 88, 96, 259, 266, 277] [69, 89, 97, 260, 267, 278] [70, 98, 111, 261, 268, 300] [71, 91, 99, 262, 269, 301] [72, 92, 100, 263, 270, 302] [73, 93, 101, 264, 271, 282] [74, 94, 102, 265, 272, 283] [75, 95, 103, 266, 273, 284] [76, 96, 104, 267, 274, 285] [77, 105, 118, 268, 275, 307] [78, 98, 106, 269, 276, 308] [79, 99, 107, 270, 277, 309] [80, 100, 108, 271, 278, 289] [81, 101, 109, 272, 279, 290] [82, 102, 110, 273, 280, 291] [83, 103, 111, 274, 281, 292] [84, 112, 125, 275, 282, 314] [85, 105, 113, 276, 283, 315] [86, 106, 114, 277, 284, 316] [87, 107, 115, 278, 285, 296] [88, 108, 116, 279, 286, 297] [89, 109, 117, 280, 287, 298] [90, 110, 118, 281, 288, 299] [91, 119, 132, 282, 289, 321] [92, 112, 120, 283, 290, 322] [93, 113, 121, 284, 291, 323] [94, 114, 122, 285, 292, 303] [95, 115, 123, 286, 293, 304] [96, 116, 124, 287, 294, 305] [97, 117, 125, 288, 295, 306] [98, 126, 139, 289, 296, 327] [99, 119, 127, 290, 297, 328] [100, 120, 128, 291, 298, 329] [101, 121, 129, 292, 299, 310] [102, 122, 130, 293, 300, 311] [103, 123, 131, 294, 301, 312] [104, 124, 132, 295, 302, 313] [105, 133, 146, 296, 303, 332] [106, 126, 134, 297, 304, 333] [107, 127, 135, 170, 298, 305] [108, 128, 136, 299, 306, 317] [109, 129, 137, 300, 307, 318] [110, 130, 138, 301, 308, 319] [111, 131, 139, 302, 309, 320] [112, 140, 153, 303, 310, 335] [113, 133, 141, 304, 311, 336] [114, 134, 142, 171, 305, 312] [115, 135, 143, 172, 306, 313] [116, 136, 144, 307, 314, 324] [117, 137, 145, 308, 315, 325] [118, 138, 146, 309, 316, 326] [119, 147, 159, 310, 317, 337] [120, 140, 148, 311, 318, 338] [121, 141, 149, 173, 312, 319] [122, 142, 150, 174, 313, 320] [123, 143, 151, 175, 314, 321] [124, 144, 152, 315, 322, 330] [125, 145, 153, 316, 323, 331] [127, 147, 154, 318, 324, 339] [128, 148, 155, 176, 319, 325] [129, 149, 156, 177, 320, 326] [130, 150, 157, 178, 321, 327] [131, 151, 158, 179, 322, 328] [132, 152, 159, 323, 329, 334] [135, 154, 160, 180, 325, 330] [136, 155, 161, 181, 326, 331] [137, 156, 162, 182, 327, 332] [138, 157, 163, 183, 328, 333] [143, 160, 164, 186, 331, 334] [144, 161, 165, 187, 332, 335] [145, 162, 166, 188, 333, 336] [151, 164, 167, 193, 335, 337] [152, 165, 168, 194, 336, 338] [158, 167, 169, 201, 338, 339]
raw submission JSON