[[360,12,24]] · QEC Challenge

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[[360,12,24]] d ≤

360

kd²/n

19.2

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Distance

d_X 24 · witness weight 24 (claimed upper_bound)

witness operator (support, 24 qubits)

[63, 65, 74, 75, 87, 109, 114, 135, 146, 147, 158, 171, 173, 189, 254, 264, 265, 288, 290, 313, 326, 335, 336, 352]

d_Z 24 · witness weight 24 (claimed upper_bound)

witness operator (support, 24 qubits)

[37, 46, 49, 62, 70, 73, 84, 107, 119, 126, 130, 141, 142, 162, 243, 264, 273, 275, 285, 296, 308, 319, 321, 322]

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^-1y³, g(x,y)=1+y+x³y^-1 on the abelian quotient group Z²/L with twist basis a_1=[0, 30], a_2=[6, 6] (n=2|det[a_1,a_2]|=2*180). 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 180 · Z-checks 180

H_X (180 checks, sparse supports)

[0, 1, 108, 180, 182, 287] [1, 3, 9, 181, 184, 299] [2, 4, 120, 182, 185, 186] [3, 6, 13, 183, 187, 311] [4, 7, 14, 184, 188, 190] [5, 8, 132, 185, 189, 191] [6, 10, 18, 186, 191, 323] [7, 11, 19, 187, 192, 195] [8, 12, 20, 188, 193, 196] [9, 13, 144, 189, 194, 197] [10, 15, 24, 190, 196, 334] [11, 16, 25, 191, 197, 201] [12, 17, 26, 192, 198, 202] [13, 18, 27, 193, 199, 203] [14, 19, 155, 194, 200, 204] [15, 21, 31, 195, 202, 343] [16, 22, 32, 196, 203, 208] [17, 23, 33, 197, 204, 209] [18, 24, 34, 198, 205, 210] [19, 25, 35, 199, 206, 211] [20, 26, 164, 200, 207, 212] [21, 28, 39, 201, 209, 350] [22, 29, 40, 202, 210, 216] [23, 30, 41, 203, 211, 217] [24, 31, 42, 204, 212, 218] [25, 32, 43, 205, 213, 219] [26, 33, 44, 206, 214, 220] [27, 34, 171, 207, 215, 221] [28, 36, 48, 208, 217, 355] [29, 37, 49, 209, 218, 225] [30, 38, 50, 210, 219, 226] [31, 39, 51, 211, 220, 227] [32, 40, 52, 212, 221, 228] [33, 41, 53, 213, 222, 229] [34, 42, 54, 214, 223, 230] [35, 43, 176, 215, 224, 231] [36, 45, 58, 216, 226, 358] [37, 46, 59, 217, 227, 235] [38, 47, 60, 218, 228, 236] [39, 48, 61, 219, 229, 237] [40, 49, 62, 220, 230, 238] [41, 50, 63, 221, 231, 239] [42, 51, 64, 222, 232, 240] [43, 52, 65, 223, 233, 241] [44, 53, 179, 224, 234, 242] [45, 55, 69, 225, 236, 329] [46, 56, 70, 226, 237, 246] [47, 57, 71, 227, 238, 247] [48, 58, 72, 228, 239, 248] [49, 59, 73, 229, 240, 249] [50, 60, 74, 230, 241, 250] [51, 61, 75, 231, 242, 251] [52, 62, 76, 232, 243, 252] [53, 63, 77, 233, 244, 253] [54, 64, 138, 234, 245, 254] [55, 66, 81, 235, 247, 339] [56, 67, 82, 236, 248, 258] [57, 68, 83, 237, 249, 259] [0, 58, 69, 238, 250, 260] [59, 70, 84, 239, 251, 261] [60, 71, 85, 240, 252, 262] [61, 72, 86, 241, 253, 263] [62, 73, 87, 180, 242, 254] [63, 74, 88, 243, 255, 264] [64, 75, 89, 244, 256, 265] [65, 76, 150, 245, 257, 266] [66, 78, 93, 246, 259, 347] [67, 79, 94, 247, 260, 270] [68, 80, 95, 248, 261, 271] [1, 69, 81, 249, 262, 272] [2, 70, 82, 250, 263, 273] [71, 83, 96, 180, 251, 274] [0, 72, 97, 252, 264, 275] [73, 84, 98, 181, 253, 265] [74, 85, 99, 182, 254, 266] [75, 86, 100, 255, 267, 276] [76, 87, 101, 256, 268, 277] [77, 88, 160, 257, 269, 278] [78, 90, 105, 258, 271, 353] [79, 91, 106, 259, 272, 282] [80, 92, 107, 260, 273, 283] [3, 81, 93, 261, 274, 284] [4, 82, 94, 262, 275, 285] [5, 83, 95, 181, 263, 286] [2, 84, 109, 183, 264, 276] [85, 96, 110, 184, 265, 277] [86, 97, 111, 185, 266, 278] [87, 98, 112, 267, 279, 288] [88, 99, 113, 268, 280, 289] [89, 100, 168, 269, 281, 290] [90, 102, 117, 270, 283, 357] [91, 103, 118, 271, 284, 294] [92, 104, 119, 272, 285, 295] [6, 93, 105, 273, 286, 296] [7, 94, 106, 274, 287, 297] [8, 95, 107, 183, 275, 298] [5, 96, 121, 187, 276, 288] [97, 108, 122, 188, 277, 289] [98, 109, 123, 189, 278, 290] [99, 110, 124, 279, 291, 300] [100, 111, 125, 280, 292, 301] [101, 112, 174, 281, 293, 302] [102, 114, 129, 282, 295, 359] [103, 115, 130, 283, 296, 306] [104, 116, 131, 284, 297, 307] [10, 105, 117, 285, 298, 308] [11, 106, 118, 286, 299, 309] [12, 107, 119, 186, 287, 310] [9, 108, 133, 192, 288, 300] [109, 120, 134, 193, 289, 301] [110, 121, 135, 194, 290, 302] [111, 122, 136, 291, 303, 312] [112, 123, 137, 292, 304, 313] [113, 124, 178, 293, 305, 314] [114, 126, 141, 257, 294, 307] [115, 127, 142, 295, 308, 318] [116, 128, 143, 296, 309, 319] [15, 117, 129, 297, 310, 320] [16, 118, 130, 298, 311, 321] [17, 119, 131, 190, 299, 322] [14, 120, 145, 198, 300, 312] [121, 132, 146, 199, 301, 313] [122, 133, 147, 200, 302, 314] [123, 134, 148, 303, 315, 324] [124, 135, 149, 304, 316, 325] [66, 125, 136, 305, 317, 326] [126, 138, 152, 268, 306, 319] [127, 139, 153, 269, 307, 320] [128, 140, 154, 308, 321, 330] [21, 129, 141, 309, 322, 331] [22, 130, 142, 310, 323, 332] [23, 131, 143, 195, 311, 333] [20, 132, 156, 205, 312, 324] [133, 144, 157, 206, 313, 325] [134, 145, 158, 207, 314, 326] [135, 146, 159, 315, 327, 335] [78, 136, 147, 316, 328, 336] [79, 137, 148, 317, 329, 337] [89, 138, 161, 279, 318, 330] [139, 150, 162, 280, 319, 331] [140, 151, 163, 281, 320, 332] [28, 141, 152, 321, 333, 340] [29, 142, 153, 322, 334, 341] [30, 143, 154, 201, 323, 342] [27, 144, 165, 213, 324, 335] [145, 155, 166, 214, 325, 336] [146, 156, 167, 215, 326, 337] [90, 147, 157, 327, 338, 344] [91, 148, 158, 328, 339, 345] [92, 149, 159, 258, 329, 346] [101, 150, 169, 291, 330, 340] [151, 160, 170, 292, 331, 341] [36, 152, 161, 293, 332, 342] [37, 153, 162, 333, 343, 348] [38, 154, 163, 208, 334, 349] [35, 155, 172, 222, 335, 344] [156, 164, 173, 223, 336, 345] [102, 157, 165, 224, 337, 346] [103, 158, 166, 338, 347, 351] [104, 159, 167, 270, 339, 352] [113, 160, 175, 303, 340, 348] [45, 161, 168, 304, 341, 349] [46, 162, 169, 305, 342, 350] [47, 163, 170, 216, 343, 354] [44, 164, 177, 232, 344, 351] [114, 165, 171, 233, 345, 352] [115, 166, 172, 234, 346, 353] [116, 167, 173, 282, 347, 356] [55, 125, 168, 315, 348, 354] [56, 169, 174, 316, 349, 355] [57, 170, 175, 225, 317, 350] [54, 126, 171, 243, 351, 356] [127, 172, 176, 244, 352, 357] [128, 173, 177, 245, 294, 353] [67, 137, 174, 327, 354, 358] [68, 175, 178, 235, 328, 355] [65, 139, 176, 255, 356, 359] [140, 177, 179, 256, 306, 357] [80, 149, 178, 246, 338, 358] [77, 151, 179, 267, 318, 359]

H_Z (180 checks, sparse supports)

[0, 62, 71, 180, 238, 252] [1, 73, 83, 180, 181, 249] [0, 2, 74, 182, 250, 264] [3, 84, 95, 181, 183, 261] [1, 4, 85, 182, 184, 262] [2, 5, 86, 185, 263, 276] [2, 6, 107, 183, 186, 273] [3, 7, 96, 184, 187, 274] [4, 8, 97, 185, 188, 275] [5, 9, 98, 181, 189, 288] [4, 10, 119, 186, 190, 285] [5, 6, 11, 187, 191, 286] [7, 12, 108, 188, 192, 287] [8, 13, 109, 183, 189, 193] [9, 14, 110, 184, 194, 300] [7, 15, 131, 190, 195, 297] [8, 10, 16, 191, 196, 298] [9, 11, 17, 192, 197, 299] [12, 18, 120, 186, 193, 198] [13, 19, 121, 187, 194, 199] [14, 20, 122, 188, 200, 312] [11, 21, 143, 195, 201, 309] [12, 15, 22, 196, 202, 310] [13, 16, 23, 197, 203, 311] [14, 17, 24, 190, 198, 204] [18, 25, 132, 191, 199, 205] [19, 26, 133, 192, 200, 206] [20, 27, 134, 193, 207, 324] [16, 28, 154, 201, 208, 321] [17, 21, 29, 202, 209, 322] [18, 22, 30, 203, 210, 323] [19, 23, 31, 195, 204, 211] [20, 24, 32, 196, 205, 212] [25, 33, 144, 197, 206, 213] [26, 34, 145, 198, 207, 214] [27, 35, 146, 199, 215, 335] [22, 36, 163, 208, 216, 332] [23, 28, 37, 209, 217, 333] [24, 29, 38, 210, 218, 334] [25, 30, 39, 201, 211, 219] [26, 31, 40, 202, 212, 220] [27, 32, 41, 203, 213, 221] [33, 42, 155, 204, 214, 222] [34, 43, 156, 205, 215, 223] [35, 44, 157, 206, 224, 344] [29, 45, 170, 216, 225, 341] [30, 36, 46, 217, 226, 342] [31, 37, 47, 218, 227, 343] [32, 38, 48, 208, 219, 228] [33, 39, 49, 209, 220, 229] [34, 40, 50, 210, 221, 230] [35, 41, 51, 211, 222, 231] [42, 52, 164, 212, 223, 232] [43, 53, 165, 213, 224, 233] [44, 54, 166, 214, 234, 351] [37, 55, 175, 225, 235, 348] [38, 45, 56, 226, 236, 349] [39, 46, 57, 227, 237, 350] [40, 47, 58, 216, 228, 238] [41, 48, 59, 217, 229, 239] [42, 49, 60, 218, 230, 240] [43, 50, 61, 219, 231, 241] [44, 51, 62, 220, 232, 242] [52, 63, 171, 221, 233, 243] [53, 64, 172, 222, 234, 244] [54, 65, 173, 223, 245, 356] [46, 66, 178, 235, 246, 305] [47, 55, 67, 236, 247, 354] [48, 56, 68, 237, 248, 355] [49, 57, 69, 225, 238, 249] [50, 58, 70, 226, 239, 250] [51, 59, 71, 227, 240, 251] [52, 60, 72, 228, 241, 252] [53, 61, 73, 229, 242, 253] [54, 62, 74, 230, 243, 254] [63, 75, 176, 231, 244, 255] [64, 76, 177, 232, 245, 256] [65, 77, 114, 233, 257, 359] [56, 78, 149, 246, 258, 316] [57, 66, 79, 247, 259, 317] [58, 67, 80, 248, 260, 358] [59, 68, 81, 235, 249, 261] [60, 69, 82, 236, 250, 262] [61, 70, 83, 237, 251, 263] [63, 72, 84, 239, 253, 264] [64, 73, 85, 240, 254, 265] [65, 74, 86, 241, 255, 266] [75, 87, 179, 242, 256, 267] [76, 88, 126, 243, 257, 268] [77, 89, 127, 244, 269, 318] [67, 90, 159, 258, 270, 327] [68, 78, 91, 259, 271, 328] [69, 79, 92, 260, 272, 329] [70, 80, 93, 246, 261, 273] [71, 81, 94, 247, 262, 274] [72, 82, 95, 248, 263, 275] [75, 84, 96, 251, 265, 276] [76, 85, 97, 252, 266, 277] [77, 86, 98, 253, 267, 278] [87, 99, 138, 254, 268, 279] [88, 100, 139, 255, 269, 280] [89, 101, 140, 256, 281, 330] [79, 102, 167, 270, 282, 337] [80, 90, 103, 271, 283, 338] [81, 91, 104, 272, 284, 339] [82, 92, 105, 258, 273, 285] [83, 93, 106, 259, 274, 286] [0, 94, 107, 260, 275, 287] [87, 96, 108, 180, 277, 288] [88, 97, 109, 264, 278, 289] [89, 98, 110, 265, 279, 290] [99, 111, 150, 266, 280, 291] [100, 112, 151, 267, 281, 292] [101, 113, 152, 268, 293, 340] [91, 114, 173, 282, 294, 345] [92, 102, 115, 283, 295, 346] [93, 103, 116, 284, 296, 347] [94, 104, 117, 270, 285, 297] [95, 105, 118, 271, 286, 298] [1, 106, 119, 272, 287, 299] [99, 108, 120, 182, 289, 300] [100, 109, 121, 276, 290, 301] [101, 110, 122, 277, 291, 302] [111, 123, 160, 278, 292, 303] [112, 124, 161, 279, 293, 304] [113, 125, 162, 280, 305, 348] [103, 126, 177, 294, 306, 351] [104, 114, 127, 295, 307, 352] [105, 115, 128, 296, 308, 353] [106, 116, 129, 282, 297, 309] [107, 117, 130, 283, 298, 310] [3, 118, 131, 284, 299, 311] [111, 120, 132, 185, 301, 312] [112, 121, 133, 288, 302, 313] [113, 122, 134, 289, 303, 314] [123, 135, 168, 290, 304, 315] [124, 136, 169, 291, 305, 316] [125, 137, 170, 292, 317, 354] [115, 138, 179, 234, 306, 318] [116, 126, 139, 307, 319, 356] [117, 127, 140, 308, 320, 357] [118, 128, 141, 294, 309, 321] [119, 129, 142, 295, 310, 322] [6, 130, 143, 296, 311, 323] [123, 132, 144, 189, 313, 324] [124, 133, 145, 300, 314, 325] [125, 134, 146, 301, 315, 326] [135, 147, 174, 302, 316, 327] [136, 148, 175, 303, 317, 328] [45, 137, 149, 304, 329, 358] [128, 138, 150, 245, 319, 330] [129, 139, 151, 320, 331, 359] [130, 140, 152, 306, 321, 332] [131, 141, 153, 307, 322, 333] [10, 142, 154, 308, 323, 334] [135, 144, 155, 194, 325, 335] [136, 145, 156, 312, 326, 336] [137, 146, 157, 313, 327, 337] [147, 158, 178, 314, 328, 338] [55, 148, 159, 315, 329, 339] [141, 150, 160, 257, 331, 340] [142, 151, 161, 318, 332, 341] [143, 152, 162, 319, 333, 342] [15, 153, 163, 320, 334, 343] [147, 155, 164, 200, 336, 344] [148, 156, 165, 324, 337, 345] [149, 157, 166, 325, 338, 346] [66, 158, 167, 326, 339, 347] [153, 160, 168, 269, 341, 348] [154, 161, 169, 330, 342, 349] [21, 162, 170, 331, 343, 350] [158, 164, 171, 207, 345, 351] [159, 165, 172, 335, 346, 352] [78, 166, 173, 336, 347, 353] [163, 168, 174, 281, 349, 354] [28, 169, 175, 340, 350, 355] [167, 171, 176, 215, 352, 356] [90, 172, 177, 344, 353, 357] [36, 174, 178, 293, 355, 358] [102, 176, 179, 224, 357, 359]

raw submission JSON