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[[350,6,26]] d ≤
n
350
k
6
d
26
kd²/n
11.589
w
6

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Distance

d_X 26 · witness weight 26 (claimed upper_bound)
witness operator (support, 26 qubits)
[43, 44, 53, 73, 102, 128, 147, 158, 171, 172, 185, 213, 229, 258, 287, 299, 303, 308, 312, 316, 320, 323, 327, 331, 340, 349]
d_Z 28 · witness weight 28 (claimed upper_bound)
witness operator (support, 28 qubits)
[15, 21, 56, 64, 85, 93, 94, 102, 104, 122, 125, 131, 132, 134, 141, 142, 152, 175, 185, 251, 272, 278, 280, 281, 282, 288, 291, 292]
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+x2y2, g(x,y)=1+y+x-4y on the abelian quotient group Z2/L with twist basis a_1=[0, 35], a_2=[5, 13] (n=2|det[a_1,a_2]|=2*175). 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 175 · Z-checks 175
H_X (175 checks, sparse supports)
[0, 1, 12, 175, 177, 288] [1, 3, 17, 176, 179, 297] [2, 4, 18, 177, 180, 298] [3, 6, 23, 178, 182, 306] [4, 7, 24, 179, 183, 307] [5, 8, 25, 180, 184, 308] [6, 10, 30, 181, 186, 314] [7, 11, 31, 182, 187, 315] [8, 12, 32, 183, 188, 316] [9, 13, 33, 184, 189, 317] [10, 15, 38, 177, 185, 191] [11, 16, 39, 186, 192, 322] [12, 17, 40, 187, 193, 323] [13, 18, 41, 188, 194, 324] [14, 19, 42, 189, 195, 325] [15, 21, 47, 179, 190, 197] [16, 22, 48, 180, 191, 198] [17, 23, 49, 192, 199, 329] [18, 24, 50, 193, 200, 330] [19, 25, 51, 194, 201, 331] [20, 26, 52, 195, 202, 332] [21, 28, 56, 182, 196, 204] [22, 29, 57, 183, 197, 205] [23, 30, 58, 184, 198, 206] [24, 31, 59, 199, 207, 335] [25, 32, 60, 200, 208, 336] [26, 33, 61, 201, 209, 337] [27, 34, 62, 202, 210, 338] [28, 36, 65, 186, 203, 212] [29, 37, 66, 187, 204, 213] [30, 38, 67, 188, 205, 214] [31, 39, 68, 189, 206, 215] [32, 40, 69, 207, 216, 340] [33, 41, 70, 208, 217, 341] [34, 42, 71, 209, 218, 342] [35, 43, 72, 210, 219, 343] [36, 45, 74, 191, 211, 221] [37, 46, 75, 192, 212, 222] [38, 47, 76, 193, 213, 223] [39, 48, 77, 194, 214, 224] [40, 49, 78, 195, 215, 225] [41, 50, 79, 216, 226, 344] [42, 51, 80, 217, 227, 345] [43, 52, 81, 218, 228, 346] [44, 53, 82, 196, 219, 229] [45, 54, 83, 197, 220, 230] [46, 55, 84, 198, 221, 231] [47, 56, 85, 199, 222, 232] [48, 57, 86, 200, 223, 233] [49, 58, 87, 201, 224, 234] [50, 59, 88, 202, 225, 235] [51, 60, 89, 226, 236, 347] [52, 61, 90, 227, 237, 348] [53, 62, 91, 203, 228, 238] [54, 63, 92, 204, 229, 239] [55, 64, 93, 205, 230, 240] [56, 65, 94, 206, 231, 241] [57, 66, 95, 207, 232, 242] [58, 67, 96, 208, 233, 243] [59, 68, 97, 209, 234, 244] [60, 69, 98, 210, 235, 245] [61, 70, 99, 236, 246, 349] [62, 71, 100, 211, 237, 247] [63, 72, 101, 212, 238, 248] [64, 73, 102, 213, 239, 249] [65, 74, 103, 214, 240, 250] [66, 75, 104, 215, 241, 251] [67, 76, 105, 216, 242, 252] [68, 77, 106, 217, 243, 253] [69, 78, 107, 218, 244, 254] [70, 79, 108, 219, 245, 255] [71, 80, 109, 220, 246, 256] [72, 81, 110, 221, 247, 257] [73, 82, 111, 222, 248, 258] [74, 83, 112, 223, 249, 259] [75, 84, 113, 224, 250, 260] [76, 85, 114, 225, 251, 261] [77, 86, 115, 226, 252, 262] [78, 87, 116, 227, 253, 263] [79, 88, 117, 228, 254, 264] [80, 89, 118, 229, 255, 265] [81, 90, 119, 230, 256, 266] [82, 91, 120, 231, 257, 267] [83, 92, 121, 232, 258, 268] [84, 93, 122, 233, 259, 269] [85, 94, 123, 234, 260, 270] [86, 95, 124, 235, 261, 271] [87, 96, 125, 236, 262, 272] [88, 97, 126, 237, 263, 273] [89, 98, 127, 238, 264, 274] [90, 99, 128, 239, 265, 275] [91, 100, 129, 240, 266, 276] [92, 101, 130, 241, 267, 277] [93, 102, 131, 242, 268, 278] [94, 103, 132, 243, 269, 279] [95, 104, 133, 244, 270, 280] [96, 105, 134, 245, 271, 281] [97, 106, 135, 246, 272, 282] [98, 107, 136, 247, 273, 283] [99, 108, 137, 248, 274, 284] [100, 109, 138, 249, 275, 285] [0, 101, 110, 250, 276, 286] [102, 111, 139, 251, 277, 287] [103, 112, 140, 252, 278, 288] [104, 113, 141, 253, 279, 289] [105, 114, 142, 254, 280, 290] [106, 115, 143, 255, 281, 291] [107, 116, 144, 256, 282, 292] [108, 117, 145, 257, 283, 293] [109, 118, 146, 258, 284, 294] [1, 110, 119, 259, 285, 295] [2, 111, 120, 260, 286, 296] [112, 121, 147, 261, 287, 297] [113, 122, 148, 262, 288, 298] [114, 123, 149, 263, 289, 299] [115, 124, 150, 264, 290, 300] [116, 125, 151, 265, 291, 301] [117, 126, 152, 266, 292, 302] [118, 127, 153, 267, 293, 303] [3, 119, 128, 268, 294, 304] [4, 120, 129, 269, 295, 305] [5, 121, 130, 270, 296, 306] [122, 131, 154, 271, 297, 307] [123, 132, 155, 272, 298, 308] [124, 133, 156, 273, 299, 309] [125, 134, 157, 274, 300, 310] [126, 135, 158, 275, 301, 311] [127, 136, 159, 276, 302, 312] [6, 128, 137, 277, 303, 313] [7, 129, 138, 175, 278, 304] [0, 8, 130, 279, 305, 314] [9, 131, 139, 280, 306, 315] [132, 140, 160, 281, 307, 316] [133, 141, 161, 282, 308, 317] [134, 142, 162, 283, 309, 318] [135, 143, 163, 284, 310, 319] [136, 144, 164, 285, 311, 320] [10, 137, 145, 286, 312, 321] [11, 138, 146, 176, 287, 313] [2, 13, 139, 289, 314, 322] [14, 140, 147, 290, 315, 323] [141, 148, 165, 291, 316, 324] [142, 149, 166, 292, 317, 325] [143, 150, 167, 293, 318, 326] [144, 151, 168, 294, 319, 327] [15, 145, 152, 295, 320, 328] [16, 146, 153, 178, 296, 321] [5, 19, 147, 299, 322, 329] [20, 148, 154, 300, 323, 330] [149, 155, 169, 301, 324, 331] [150, 156, 170, 302, 325, 332] [151, 157, 171, 303, 326, 333] [21, 152, 158, 304, 327, 334] [22, 153, 159, 181, 305, 328] [9, 26, 154, 309, 329, 335] [27, 155, 160, 310, 330, 336] [156, 161, 172, 311, 331, 337] [157, 162, 173, 312, 332, 338] [28, 158, 163, 313, 333, 339] [29, 159, 164, 175, 185, 334] [14, 34, 160, 318, 335, 340] [35, 161, 165, 319, 336, 341] [162, 166, 174, 320, 337, 342] [36, 163, 167, 321, 338, 343] [37, 164, 168, 176, 190, 339] [20, 43, 165, 326, 340, 344] [44, 166, 169, 327, 341, 345] [45, 167, 170, 328, 342, 346] [46, 168, 171, 178, 196, 343] [27, 53, 169, 333, 344, 347] [54, 170, 172, 334, 345, 348] [55, 171, 173, 181, 203, 346] [35, 63, 172, 339, 347, 349] [64, 173, 174, 185, 211, 348] [44, 73, 174, 190, 220, 349]
H_Z (175 checks, sparse supports)
[0, 129, 159, 175, 276, 305] [1, 138, 164, 175, 176, 285] [0, 2, 10, 177, 286, 314] [3, 146, 168, 176, 178, 294] [1, 4, 15, 177, 179, 295] [2, 5, 16, 180, 296, 322] [6, 153, 171, 178, 181, 303] [3, 7, 21, 179, 182, 304] [4, 8, 22, 180, 183, 305] [5, 9, 23, 184, 306, 329] [10, 159, 173, 181, 185, 312] [6, 11, 28, 182, 186, 313] [7, 12, 29, 175, 183, 187] [8, 13, 30, 184, 188, 314] [9, 14, 31, 189, 315, 335] [15, 164, 174, 185, 190, 320] [10, 16, 36, 186, 191, 321] [11, 17, 37, 176, 187, 192] [12, 18, 38, 177, 188, 193] [13, 19, 39, 189, 194, 322] [14, 20, 40, 195, 323, 340] [21, 44, 168, 190, 196, 327] [15, 22, 45, 191, 197, 328] [16, 23, 46, 178, 192, 198] [17, 24, 47, 179, 193, 199] [18, 25, 48, 180, 194, 200] [19, 26, 49, 195, 201, 329] [20, 27, 50, 202, 330, 344] [28, 53, 171, 196, 203, 333] [21, 29, 54, 197, 204, 334] [22, 30, 55, 181, 198, 205] [23, 31, 56, 182, 199, 206] [24, 32, 57, 183, 200, 207] [25, 33, 58, 184, 201, 208] [26, 34, 59, 202, 209, 335] [27, 35, 60, 210, 336, 347] [36, 62, 173, 203, 211, 338] [28, 37, 63, 204, 212, 339] [29, 38, 64, 185, 205, 213] [30, 39, 65, 186, 206, 214] [31, 40, 66, 187, 207, 215] [32, 41, 67, 188, 208, 216] [33, 42, 68, 189, 209, 217] [34, 43, 69, 210, 218, 340] [35, 44, 70, 219, 341, 349] [45, 71, 174, 211, 220, 342] [36, 46, 72, 212, 221, 343] [37, 47, 73, 190, 213, 222] [38, 48, 74, 191, 214, 223] [39, 49, 75, 192, 215, 224] [40, 50, 76, 193, 216, 225] [41, 51, 77, 194, 217, 226] [42, 52, 78, 195, 218, 227] [43, 53, 79, 219, 228, 344] [44, 54, 80, 220, 229, 345] [45, 55, 81, 221, 230, 346] [46, 56, 82, 196, 222, 231] [47, 57, 83, 197, 223, 232] [48, 58, 84, 198, 224, 233] [49, 59, 85, 199, 225, 234] [50, 60, 86, 200, 226, 235] [51, 61, 87, 201, 227, 236] [52, 62, 88, 202, 228, 237] [53, 63, 89, 229, 238, 347] [54, 64, 90, 230, 239, 348] [55, 65, 91, 203, 231, 240] [56, 66, 92, 204, 232, 241] [57, 67, 93, 205, 233, 242] [58, 68, 94, 206, 234, 243] [59, 69, 95, 207, 235, 244] [60, 70, 96, 208, 236, 245] [61, 71, 97, 209, 237, 246] [62, 72, 98, 210, 238, 247] [63, 73, 99, 239, 248, 349] [64, 74, 100, 211, 240, 249] [65, 75, 101, 212, 241, 250] [66, 76, 102, 213, 242, 251] [67, 77, 103, 214, 243, 252] [68, 78, 104, 215, 244, 253] [69, 79, 105, 216, 245, 254] [70, 80, 106, 217, 246, 255] [71, 81, 107, 218, 247, 256] [72, 82, 108, 219, 248, 257] [73, 83, 109, 220, 249, 258] [74, 84, 110, 221, 250, 259] [75, 85, 111, 222, 251, 260] [76, 86, 112, 223, 252, 261] [77, 87, 113, 224, 253, 262] [78, 88, 114, 225, 254, 263] [79, 89, 115, 226, 255, 264] [80, 90, 116, 227, 256, 265] [81, 91, 117, 228, 257, 266] [82, 92, 118, 229, 258, 267] [83, 93, 119, 230, 259, 268] [84, 94, 120, 231, 260, 269] [85, 95, 121, 232, 261, 270] [86, 96, 122, 233, 262, 271] [87, 97, 123, 234, 263, 272] [88, 98, 124, 235, 264, 273] [89, 99, 125, 236, 265, 274] [90, 100, 126, 237, 266, 275] [91, 101, 127, 238, 267, 276] [92, 102, 128, 239, 268, 277] [93, 103, 129, 240, 269, 278] [94, 104, 130, 241, 270, 279] [95, 105, 131, 242, 271, 280] [96, 106, 132, 243, 272, 281] [97, 107, 133, 244, 273, 282] [98, 108, 134, 245, 274, 283] [99, 109, 135, 246, 275, 284] [100, 110, 136, 247, 276, 285] [101, 111, 137, 248, 277, 286] [102, 112, 138, 249, 278, 287] [0, 103, 113, 250, 279, 288] [104, 114, 139, 251, 280, 289] [105, 115, 140, 252, 281, 290] [106, 116, 141, 253, 282, 291] [107, 117, 142, 254, 283, 292] [108, 118, 143, 255, 284, 293] [109, 119, 144, 256, 285, 294] [110, 120, 145, 257, 286, 295] [111, 121, 146, 258, 287, 296] [1, 112, 122, 259, 288, 297] [2, 113, 123, 260, 289, 298] [114, 124, 147, 261, 290, 299] [115, 125, 148, 262, 291, 300] [116, 126, 149, 263, 292, 301] [117, 127, 150, 264, 293, 302] [118, 128, 151, 265, 294, 303] [119, 129, 152, 266, 295, 304] [120, 130, 153, 267, 296, 305] [3, 121, 131, 268, 297, 306] [4, 122, 132, 269, 298, 307] [5, 123, 133, 270, 299, 308] [124, 134, 154, 271, 300, 309] [125, 135, 155, 272, 301, 310] [126, 136, 156, 273, 302, 311] [127, 137, 157, 274, 303, 312] [128, 138, 158, 275, 304, 313] [6, 130, 139, 277, 306, 314] [7, 131, 140, 278, 307, 315] [8, 132, 141, 279, 308, 316] [9, 133, 142, 280, 309, 317] [134, 143, 160, 281, 310, 318] [135, 144, 161, 282, 311, 319] [136, 145, 162, 283, 312, 320] [137, 146, 163, 284, 313, 321] [11, 139, 147, 287, 315, 322] [12, 140, 148, 288, 316, 323] [13, 141, 149, 289, 317, 324] [14, 142, 150, 290, 318, 325] [143, 151, 165, 291, 319, 326] [144, 152, 166, 292, 320, 327] [145, 153, 167, 293, 321, 328] [17, 147, 154, 297, 323, 329] [18, 148, 155, 298, 324, 330] [19, 149, 156, 299, 325, 331] [20, 150, 157, 300, 326, 332] [151, 158, 169, 301, 327, 333] [152, 159, 170, 302, 328, 334] [24, 154, 160, 307, 330, 335] [25, 155, 161, 308, 331, 336] [26, 156, 162, 309, 332, 337] [27, 157, 163, 310, 333, 338] [158, 164, 172, 311, 334, 339] [32, 160, 165, 316, 336, 340] [33, 161, 166, 317, 337, 341] [34, 162, 167, 318, 338, 342] [35, 163, 168, 319, 339, 343] [41, 165, 169, 324, 341, 344] [42, 166, 170, 325, 342, 345] [43, 167, 171, 326, 343, 346] [51, 169, 172, 331, 345, 347] [52, 170, 173, 332, 346, 348] [61, 172, 174, 337, 348, 349]
raw submission JSON