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[[308,6,24]] d ≤
n
308
k
6
d
24
kd²/n
11.221
w
6

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Distance

d_X 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[2, 63, 67, 71, 73, 75, 79, 81, 82, 85, 87, 89, 92, 94, 165, 185, 188, 193, 214, 217, 228, 236, 249, 255]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[15, 37, 44, 46, 77, 80, 88, 116, 120, 124, 140, 144, 190, 200, 220, 242, 271, 273, 280, 282, 284, 288, 289, 307]
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-1y-2, g(x,y)=1+y+x2y-1 on the abelian quotient group Z2/L with twist basis a_1=[0, 77], a_2=[2, -13] (n=2|det[a_1,a_2]|=2*154). 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 154 · Z-checks 154
H_X (154 checks, sparse supports)
[0, 1, 143, 154, 156, 243] [1, 3, 147, 155, 158, 254] [2, 4, 148, 156, 157, 159] [3, 6, 150, 157, 161, 264] [4, 7, 151, 158, 160, 162] [5, 8, 152, 159, 161, 163] [6, 10, 77, 160, 165, 273] [7, 11, 153, 161, 164, 166] [0, 8, 12, 162, 165, 167] [9, 13, 66, 163, 166, 168] [10, 15, 88, 164, 170, 281] [11, 16, 89, 165, 169, 171] [1, 12, 17, 166, 170, 172] [2, 13, 18, 167, 171, 173] [14, 19, 78, 168, 172, 174] [15, 21, 99, 169, 176, 288] [16, 22, 100, 170, 175, 177] [3, 17, 23, 171, 176, 178] [4, 18, 24, 172, 177, 179] [5, 19, 25, 173, 178, 180] [20, 26, 90, 174, 179, 181] [21, 28, 109, 175, 183, 294] [22, 29, 110, 176, 182, 184] [6, 23, 30, 177, 183, 185] [7, 24, 31, 178, 184, 186] [8, 25, 32, 179, 185, 187] [9, 26, 33, 180, 186, 188] [27, 34, 101, 181, 187, 189] [28, 36, 118, 182, 191, 299] [29, 37, 119, 183, 190, 192] [10, 30, 38, 184, 191, 193] [11, 31, 39, 185, 192, 194] [12, 32, 40, 186, 193, 195] [13, 33, 41, 187, 194, 196] [14, 34, 42, 188, 195, 197] [35, 43, 111, 189, 196, 198] [36, 45, 126, 190, 200, 303] [37, 46, 127, 191, 199, 201] [15, 38, 47, 192, 200, 202] [16, 39, 48, 193, 201, 203] [17, 40, 49, 194, 202, 204] [18, 41, 50, 195, 203, 205] [19, 42, 51, 196, 204, 206] [20, 43, 52, 197, 205, 207] [44, 53, 120, 198, 206, 208] [45, 55, 133, 199, 210, 306] [46, 56, 134, 200, 209, 211] [21, 47, 57, 201, 210, 212] [22, 48, 58, 202, 211, 213] [23, 49, 59, 203, 212, 214] [24, 50, 60, 204, 213, 215] [25, 51, 61, 205, 214, 216] [26, 52, 62, 206, 215, 217] [27, 53, 63, 207, 216, 218] [54, 64, 128, 208, 217, 219] [55, 66, 139, 154, 209, 221] [56, 67, 140, 210, 220, 222] [28, 57, 68, 211, 221, 223] [29, 58, 69, 212, 222, 224] [30, 59, 70, 213, 223, 225] [31, 60, 71, 214, 224, 226] [32, 61, 72, 215, 225, 227] [33, 62, 73, 216, 226, 228] [34, 63, 74, 217, 227, 229] [35, 64, 75, 218, 228, 230] [65, 76, 135, 219, 229, 231] [2, 66, 144, 155, 220, 232] [67, 78, 145, 156, 221, 233] [36, 68, 79, 222, 232, 234] [37, 69, 80, 223, 233, 235] [38, 70, 81, 224, 234, 236] [39, 71, 82, 225, 235, 237] [40, 72, 83, 226, 236, 238] [41, 73, 84, 227, 237, 239] [42, 74, 85, 228, 238, 240] [43, 75, 86, 229, 239, 241] [44, 76, 87, 230, 240, 242] [77, 88, 141, 231, 241, 243] [5, 78, 149, 158, 232, 244] [45, 79, 90, 159, 233, 245] [46, 80, 91, 234, 244, 246] [47, 81, 92, 235, 245, 247] [48, 82, 93, 236, 246, 248] [49, 83, 94, 237, 247, 249] [50, 84, 95, 238, 248, 250] [51, 85, 96, 239, 249, 251] [52, 86, 97, 240, 250, 252] [53, 87, 98, 241, 251, 253] [54, 88, 99, 242, 252, 254] [89, 100, 146, 157, 243, 253] [9, 55, 90, 162, 244, 255] [56, 91, 101, 163, 245, 256] [57, 92, 102, 246, 255, 257] [58, 93, 103, 247, 256, 258] [59, 94, 104, 248, 257, 259] [60, 95, 105, 249, 258, 260] [61, 96, 106, 250, 259, 261] [62, 97, 107, 251, 260, 262] [63, 98, 108, 252, 261, 263] [64, 99, 109, 253, 262, 264] [65, 100, 110, 160, 254, 263] [14, 67, 101, 167, 255, 265] [68, 102, 111, 168, 256, 266] [69, 103, 112, 257, 265, 267] [70, 104, 113, 258, 266, 268] [71, 105, 114, 259, 267, 269] [72, 106, 115, 260, 268, 270] [73, 107, 116, 261, 269, 271] [74, 108, 117, 262, 270, 272] [75, 109, 118, 263, 271, 273] [76, 110, 119, 164, 264, 272] [20, 79, 111, 173, 265, 274] [80, 112, 120, 174, 266, 275] [81, 113, 121, 267, 274, 276] [82, 114, 122, 268, 275, 277] [83, 115, 123, 269, 276, 278] [84, 116, 124, 270, 277, 279] [85, 117, 125, 271, 278, 280] [86, 118, 126, 272, 279, 281] [87, 119, 127, 169, 273, 280] [27, 91, 120, 180, 274, 282] [92, 121, 128, 181, 275, 283] [93, 122, 129, 276, 282, 284] [94, 123, 130, 277, 283, 285] [95, 124, 131, 278, 284, 286] [96, 125, 132, 279, 285, 287] [97, 126, 133, 280, 286, 288] [98, 127, 134, 175, 281, 287] [35, 102, 128, 188, 282, 289] [103, 129, 135, 189, 283, 290] [104, 130, 136, 284, 289, 291] [105, 131, 137, 285, 290, 292] [106, 132, 138, 286, 291, 293] [107, 133, 139, 287, 292, 294] [108, 134, 140, 182, 288, 293] [44, 112, 135, 197, 289, 295] [113, 136, 141, 198, 290, 296] [114, 137, 142, 291, 295, 297] [115, 138, 143, 292, 296, 298] [116, 139, 144, 293, 297, 299] [117, 140, 145, 190, 294, 298] [54, 121, 141, 207, 295, 300] [122, 142, 146, 208, 296, 301] [123, 143, 147, 297, 300, 302] [124, 144, 148, 298, 301, 303] [125, 145, 149, 199, 299, 302] [65, 129, 146, 218, 300, 304] [130, 147, 150, 219, 301, 305] [131, 148, 151, 302, 304, 306] [132, 149, 152, 209, 303, 305] [77, 136, 150, 230, 304, 307] [137, 151, 153, 154, 231, 305] [0, 138, 152, 220, 306, 307] [89, 142, 153, 155, 242, 307]
H_Z (154 checks, sparse supports)
[0, 55, 151, 154, 162, 306] [1, 66, 153, 154, 155, 166] [0, 2, 67, 156, 167, 220] [2, 3, 89, 155, 157, 171] [1, 4, 78, 156, 158, 172] [2, 5, 79, 159, 173, 232] [4, 6, 100, 157, 160, 177] [3, 5, 7, 158, 161, 178] [4, 8, 90, 159, 162, 179] [5, 9, 91, 163, 180, 244] [7, 10, 110, 160, 164, 184] [6, 8, 11, 161, 165, 185] [7, 9, 12, 162, 166, 186] [8, 13, 101, 163, 167, 187] [9, 14, 102, 168, 188, 255] [11, 15, 119, 164, 169, 192] [10, 12, 16, 165, 170, 193] [11, 13, 17, 166, 171, 194] [12, 14, 18, 167, 172, 195] [13, 19, 111, 168, 173, 196] [14, 20, 112, 174, 197, 265] [16, 21, 127, 169, 175, 201] [15, 17, 22, 170, 176, 202] [16, 18, 23, 171, 177, 203] [17, 19, 24, 172, 178, 204] [18, 20, 25, 173, 179, 205] [19, 26, 120, 174, 180, 206] [20, 27, 121, 181, 207, 274] [22, 28, 134, 175, 182, 211] [21, 23, 29, 176, 183, 212] [22, 24, 30, 177, 184, 213] [23, 25, 31, 178, 185, 214] [24, 26, 32, 179, 186, 215] [25, 27, 33, 180, 187, 216] [26, 34, 128, 181, 188, 217] [27, 35, 129, 189, 218, 282] [29, 36, 140, 182, 190, 222] [28, 30, 37, 183, 191, 223] [29, 31, 38, 184, 192, 224] [30, 32, 39, 185, 193, 225] [31, 33, 40, 186, 194, 226] [32, 34, 41, 187, 195, 227] [33, 35, 42, 188, 196, 228] [34, 43, 135, 189, 197, 229] [35, 44, 136, 198, 230, 289] [37, 45, 145, 190, 199, 233] [36, 38, 46, 191, 200, 234] [37, 39, 47, 192, 201, 235] [38, 40, 48, 193, 202, 236] [39, 41, 49, 194, 203, 237] [40, 42, 50, 195, 204, 238] [41, 43, 51, 196, 205, 239] [42, 44, 52, 197, 206, 240] [43, 53, 141, 198, 207, 241] [44, 54, 142, 208, 242, 295] [46, 55, 149, 199, 209, 244] [45, 47, 56, 200, 210, 245] [46, 48, 57, 201, 211, 246] [47, 49, 58, 202, 212, 247] [48, 50, 59, 203, 213, 248] [49, 51, 60, 204, 214, 249] [50, 52, 61, 205, 215, 250] [51, 53, 62, 206, 216, 251] [52, 54, 63, 207, 217, 252] [53, 64, 146, 208, 218, 253] [54, 65, 147, 219, 254, 300] [56, 66, 152, 163, 209, 220] [55, 57, 67, 210, 221, 255] [56, 58, 68, 211, 222, 256] [57, 59, 69, 212, 223, 257] [58, 60, 70, 213, 224, 258] [59, 61, 71, 214, 225, 259] [60, 62, 72, 215, 226, 260] [61, 63, 73, 216, 227, 261] [62, 64, 74, 217, 228, 262] [63, 65, 75, 218, 229, 263] [64, 76, 150, 219, 230, 264] [65, 77, 151, 160, 231, 304] [66, 68, 78, 168, 221, 232] [67, 69, 79, 222, 233, 265] [68, 70, 80, 223, 234, 266] [69, 71, 81, 224, 235, 267] [70, 72, 82, 225, 236, 268] [71, 73, 83, 226, 237, 269] [72, 74, 84, 227, 238, 270] [73, 75, 85, 228, 239, 271] [74, 76, 86, 229, 240, 272] [75, 77, 87, 230, 241, 273] [76, 88, 153, 164, 231, 242] [0, 77, 89, 165, 243, 307] [78, 80, 90, 174, 233, 244] [79, 81, 91, 234, 245, 274] [80, 82, 92, 235, 246, 275] [81, 83, 93, 236, 247, 276] [82, 84, 94, 237, 248, 277] [83, 85, 95, 238, 249, 278] [84, 86, 96, 239, 250, 279] [85, 87, 97, 240, 251, 280] [86, 88, 98, 241, 252, 281] [87, 89, 99, 169, 242, 253] [1, 88, 100, 170, 243, 254] [90, 92, 101, 181, 245, 255] [91, 93, 102, 246, 256, 282] [92, 94, 103, 247, 257, 283] [93, 95, 104, 248, 258, 284] [94, 96, 105, 249, 259, 285] [95, 97, 106, 250, 260, 286] [96, 98, 107, 251, 261, 287] [97, 99, 108, 252, 262, 288] [98, 100, 109, 175, 253, 263] [3, 99, 110, 176, 254, 264] [101, 103, 111, 189, 256, 265] [102, 104, 112, 257, 266, 289] [103, 105, 113, 258, 267, 290] [104, 106, 114, 259, 268, 291] [105, 107, 115, 260, 269, 292] [106, 108, 116, 261, 270, 293] [107, 109, 117, 262, 271, 294] [108, 110, 118, 182, 263, 272] [6, 109, 119, 183, 264, 273] [111, 113, 120, 198, 266, 274] [112, 114, 121, 267, 275, 295] [113, 115, 122, 268, 276, 296] [114, 116, 123, 269, 277, 297] [115, 117, 124, 270, 278, 298] [116, 118, 125, 271, 279, 299] [117, 119, 126, 190, 272, 280] [10, 118, 127, 191, 273, 281] [120, 122, 128, 208, 275, 282] [121, 123, 129, 276, 283, 300] [122, 124, 130, 277, 284, 301] [123, 125, 131, 278, 285, 302] [124, 126, 132, 279, 286, 303] [125, 127, 133, 199, 280, 287] [15, 126, 134, 200, 281, 288] [128, 130, 135, 219, 283, 289] [129, 131, 136, 284, 290, 304] [130, 132, 137, 285, 291, 305] [131, 133, 138, 286, 292, 306] [132, 134, 139, 209, 287, 293] [21, 133, 140, 210, 288, 294] [135, 137, 141, 231, 290, 295] [136, 138, 142, 291, 296, 307] [137, 139, 143, 154, 292, 297] [138, 140, 144, 220, 293, 298] [28, 139, 145, 221, 294, 299] [141, 143, 146, 243, 296, 300] [142, 144, 147, 155, 297, 301] [143, 145, 148, 156, 298, 302] [36, 144, 149, 232, 299, 303] [146, 148, 150, 157, 301, 304] [147, 149, 151, 158, 302, 305] [45, 148, 152, 159, 303, 306] [150, 152, 153, 161, 305, 307]
raw submission JSON