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[[312,8,22]] d ≤
n
312
k
8
d
22
kd²/n
12.41
w
6

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Distance

d_X 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[9, 76, 86, 97, 106, 107, 117, 126, 140, 147, 152, 154, 155, 161, 164, 183, 231, 232, 241, 272, 289, 307]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[37, 48, 57, 58, 77, 79, 88, 98, 110, 119, 143, 184, 202, 212, 223, 234, 244, 265, 291, 293, 298, 304]
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+xy3 on the abelian quotient group Z2/L with twist basis a_1=[0, 78], a_2=[2, -16] (n=2|det[a_1,a_2]|=2*156). 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 156 · Z-checks 156
H_X (156 checks, sparse supports)
[0, 1, 55, 156, 158, 169] [1, 3, 9, 157, 160, 174] [2, 4, 65, 158, 161, 175] [3, 6, 13, 159, 163, 180] [4, 7, 14, 160, 164, 181] [5, 8, 75, 161, 165, 182] [6, 10, 18, 162, 167, 187] [7, 11, 19, 163, 168, 188] [8, 12, 20, 164, 169, 189] [9, 13, 85, 165, 170, 190] [10, 15, 24, 166, 172, 195] [11, 16, 25, 167, 173, 196] [12, 17, 26, 168, 174, 197] [13, 18, 27, 169, 175, 198] [14, 19, 95, 170, 176, 199] [15, 21, 31, 171, 178, 204] [16, 22, 32, 172, 179, 205] [17, 23, 33, 173, 180, 206] [18, 24, 34, 174, 181, 207] [19, 25, 35, 175, 182, 208] [20, 26, 105, 176, 183, 209] [21, 28, 39, 177, 185, 213] [22, 29, 40, 178, 186, 214] [23, 30, 41, 179, 187, 215] [24, 31, 42, 180, 188, 216] [25, 32, 43, 181, 189, 217] [26, 33, 44, 182, 190, 218] [27, 34, 115, 183, 191, 219] [28, 36, 48, 184, 193, 222] [29, 37, 49, 185, 194, 223] [30, 38, 50, 186, 195, 224] [31, 39, 51, 187, 196, 225] [32, 40, 52, 188, 197, 226] [33, 41, 53, 189, 198, 227] [34, 42, 54, 190, 199, 228] [35, 43, 124, 191, 200, 229] [36, 45, 57, 192, 202, 231] [37, 46, 58, 193, 203, 232] [38, 47, 59, 194, 204, 233] [39, 48, 60, 195, 205, 234] [40, 49, 61, 196, 206, 235] [41, 50, 62, 197, 207, 236] [42, 51, 63, 198, 208, 237] [43, 52, 64, 199, 209, 238] [44, 53, 132, 200, 210, 239] [5, 45, 66, 176, 201, 211] [46, 55, 67, 202, 212, 241] [47, 56, 68, 203, 213, 242] [48, 57, 69, 204, 214, 243] [49, 58, 70, 205, 215, 244] [50, 59, 71, 206, 216, 245] [51, 60, 72, 207, 217, 246] [52, 61, 73, 208, 218, 247] [53, 62, 74, 209, 219, 248] [54, 63, 139, 210, 220, 249] [9, 55, 76, 183, 211, 221] [56, 65, 77, 212, 222, 251] [57, 66, 78, 213, 223, 252] [58, 67, 79, 214, 224, 253] [59, 68, 80, 215, 225, 254] [60, 69, 81, 216, 226, 255] [61, 70, 82, 217, 227, 256] [62, 71, 83, 218, 228, 257] [63, 72, 84, 219, 229, 258] [64, 73, 145, 220, 230, 259] [14, 65, 86, 191, 221, 231] [66, 75, 87, 222, 232, 261] [67, 76, 88, 223, 233, 262] [68, 77, 89, 224, 234, 263] [69, 78, 90, 225, 235, 264] [70, 79, 91, 226, 236, 265] [71, 80, 92, 227, 237, 266] [72, 81, 93, 228, 238, 267] [73, 82, 94, 229, 239, 268] [74, 83, 150, 230, 240, 269] [20, 75, 96, 200, 231, 241] [76, 85, 97, 232, 242, 271] [77, 86, 98, 233, 243, 272] [78, 87, 99, 234, 244, 273] [79, 88, 100, 235, 245, 274] [80, 89, 101, 236, 246, 275] [81, 90, 102, 237, 247, 276] [82, 91, 103, 238, 248, 277] [83, 92, 104, 239, 249, 278] [84, 93, 154, 240, 250, 279] [27, 85, 106, 210, 241, 251] [86, 95, 107, 242, 252, 280] [87, 96, 108, 243, 253, 281] [88, 97, 109, 244, 254, 282] [89, 98, 110, 245, 255, 283] [90, 99, 111, 246, 256, 284] [91, 100, 112, 247, 257, 285] [92, 101, 113, 248, 258, 286] [93, 102, 114, 249, 259, 287] [1, 94, 103, 162, 250, 260] [35, 95, 116, 220, 251, 261] [96, 105, 117, 252, 262, 288] [97, 106, 118, 253, 263, 289] [98, 107, 119, 254, 264, 290] [99, 108, 120, 255, 265, 291] [100, 109, 121, 256, 266, 292] [101, 110, 122, 257, 267, 293] [102, 111, 123, 258, 268, 294] [3, 103, 112, 166, 259, 269] [4, 104, 113, 167, 260, 270] [44, 105, 125, 230, 261, 271] [106, 115, 126, 262, 272, 295] [107, 116, 127, 263, 273, 296] [108, 117, 128, 264, 274, 297] [109, 118, 129, 265, 275, 298] [110, 119, 130, 266, 276, 299] [111, 120, 131, 267, 277, 300] [6, 112, 121, 171, 268, 278] [7, 113, 122, 172, 269, 279] [8, 114, 123, 159, 173, 270] [54, 115, 133, 240, 271, 280] [116, 124, 134, 272, 281, 301] [117, 125, 135, 273, 282, 302] [118, 126, 136, 274, 283, 303] [119, 127, 137, 275, 284, 304] [120, 128, 138, 276, 285, 305] [10, 121, 129, 177, 277, 286] [11, 122, 130, 178, 278, 287] [12, 123, 131, 162, 179, 279] [64, 124, 140, 250, 280, 288] [125, 132, 141, 281, 289, 306] [126, 133, 142, 282, 290, 307] [127, 134, 143, 283, 291, 308] [128, 135, 144, 284, 292, 309] [15, 129, 136, 184, 285, 293] [16, 130, 137, 185, 286, 294] [17, 131, 138, 166, 186, 287] [74, 132, 146, 260, 288, 295] [133, 139, 147, 289, 296, 310] [134, 140, 148, 156, 290, 297] [135, 141, 149, 291, 298, 311] [21, 136, 142, 192, 292, 299] [22, 137, 143, 193, 293, 300] [23, 138, 144, 171, 194, 294] [84, 139, 151, 270, 295, 301] [140, 145, 152, 157, 296, 302] [141, 146, 153, 158, 297, 303] [28, 142, 147, 201, 298, 304] [29, 143, 148, 202, 299, 305] [30, 144, 149, 177, 203, 300] [0, 94, 145, 159, 301, 306] [146, 150, 155, 160, 302, 307] [36, 147, 151, 161, 303, 308] [37, 148, 152, 211, 304, 309] [38, 149, 153, 184, 212, 305] [2, 104, 150, 163, 306, 310] [45, 151, 154, 156, 164, 307] [0, 46, 152, 165, 308, 311] [47, 153, 155, 192, 221, 309] [5, 114, 154, 157, 168, 310] [2, 56, 155, 170, 201, 311]
H_Z (156 checks, sparse supports)
[0, 134, 151, 156, 301, 308] [1, 140, 154, 156, 157, 250] [0, 2, 141, 158, 306, 311] [3, 114, 145, 157, 159, 259] [1, 4, 146, 158, 160, 260] [2, 5, 147, 161, 201, 310] [6, 94, 123, 159, 162, 268] [3, 7, 150, 160, 163, 269] [4, 8, 151, 161, 164, 270] [5, 9, 152, 157, 165, 211] [10, 103, 131, 162, 166, 277] [6, 11, 104, 163, 167, 278] [7, 12, 154, 164, 168, 279] [0, 8, 13, 159, 165, 169] [9, 14, 155, 160, 170, 221] [15, 112, 138, 166, 171, 285] [10, 16, 113, 167, 172, 286] [11, 17, 114, 168, 173, 287] [1, 12, 18, 162, 169, 174] [2, 13, 19, 163, 170, 175] [14, 20, 45, 164, 176, 231] [21, 121, 144, 171, 177, 292] [15, 22, 122, 172, 178, 293] [16, 23, 123, 173, 179, 294] [3, 17, 24, 166, 174, 180] [4, 18, 25, 167, 175, 181] [5, 19, 26, 168, 176, 182] [20, 27, 55, 169, 183, 241] [28, 129, 149, 177, 184, 298] [21, 29, 130, 178, 185, 299] [22, 30, 131, 179, 186, 300] [6, 23, 31, 171, 180, 187] [7, 24, 32, 172, 181, 188] [8, 25, 33, 173, 182, 189] [9, 26, 34, 174, 183, 190] [27, 35, 65, 175, 191, 251] [36, 136, 153, 184, 192, 303] [28, 37, 137, 185, 193, 304] [29, 38, 138, 186, 194, 305] [10, 30, 39, 177, 187, 195] [11, 31, 40, 178, 188, 196] [12, 32, 41, 179, 189, 197] [13, 33, 42, 180, 190, 198] [14, 34, 43, 181, 191, 199] [35, 44, 75, 182, 200, 261] [45, 142, 155, 192, 201, 307] [36, 46, 143, 193, 202, 308] [37, 47, 144, 194, 203, 309] [15, 38, 48, 184, 195, 204] [16, 39, 49, 185, 196, 205] [17, 40, 50, 186, 197, 206] [18, 41, 51, 187, 198, 207] [19, 42, 52, 188, 199, 208] [20, 43, 53, 189, 200, 209] [44, 54, 85, 190, 210, 271] [45, 55, 148, 156, 202, 211] [46, 56, 149, 203, 212, 311] [21, 47, 57, 192, 204, 213] [22, 48, 58, 193, 205, 214] [23, 49, 59, 194, 206, 215] [24, 50, 60, 195, 207, 216] [25, 51, 61, 196, 208, 217] [26, 52, 62, 197, 209, 218] [27, 53, 63, 198, 210, 219] [54, 64, 95, 199, 220, 280] [55, 65, 153, 158, 212, 221] [28, 56, 66, 201, 213, 222] [29, 57, 67, 202, 214, 223] [30, 58, 68, 203, 215, 224] [31, 59, 69, 204, 216, 225] [32, 60, 70, 205, 217, 226] [33, 61, 71, 206, 218, 227] [34, 62, 72, 207, 219, 228] [35, 63, 73, 208, 220, 229] [64, 74, 105, 209, 230, 288] [36, 65, 75, 161, 222, 231] [37, 66, 76, 211, 223, 232] [38, 67, 77, 212, 224, 233] [39, 68, 78, 213, 225, 234] [40, 69, 79, 214, 226, 235] [41, 70, 80, 215, 227, 236] [42, 71, 81, 216, 228, 237] [43, 72, 82, 217, 229, 238] [44, 73, 83, 218, 230, 239] [74, 84, 115, 219, 240, 295] [46, 75, 85, 165, 232, 241] [47, 76, 86, 221, 233, 242] [48, 77, 87, 222, 234, 243] [49, 78, 88, 223, 235, 244] [50, 79, 89, 224, 236, 245] [51, 80, 90, 225, 237, 246] [52, 81, 91, 226, 238, 247] [53, 82, 92, 227, 239, 248] [54, 83, 93, 228, 240, 249] [84, 94, 124, 229, 250, 301] [56, 85, 95, 170, 242, 251] [57, 86, 96, 231, 243, 252] [58, 87, 97, 232, 244, 253] [59, 88, 98, 233, 245, 254] [60, 89, 99, 234, 246, 255] [61, 90, 100, 235, 247, 256] [62, 91, 101, 236, 248, 257] [63, 92, 102, 237, 249, 258] [64, 93, 103, 238, 250, 259] [94, 104, 132, 239, 260, 306] [66, 95, 105, 176, 252, 261] [67, 96, 106, 241, 253, 262] [68, 97, 107, 242, 254, 263] [69, 98, 108, 243, 255, 264] [70, 99, 109, 244, 256, 265] [71, 100, 110, 245, 257, 266] [72, 101, 111, 246, 258, 267] [73, 102, 112, 247, 259, 268] [74, 103, 113, 248, 260, 269] [104, 114, 139, 249, 270, 310] [76, 105, 115, 183, 262, 271] [77, 106, 116, 251, 263, 272] [78, 107, 117, 252, 264, 273] [79, 108, 118, 253, 265, 274] [80, 109, 119, 254, 266, 275] [81, 110, 120, 255, 267, 276] [82, 111, 121, 256, 268, 277] [83, 112, 122, 257, 269, 278] [84, 113, 123, 258, 270, 279] [86, 115, 124, 191, 272, 280] [87, 116, 125, 261, 273, 281] [88, 117, 126, 262, 274, 282] [89, 118, 127, 263, 275, 283] [90, 119, 128, 264, 276, 284] [91, 120, 129, 265, 277, 285] [92, 121, 130, 266, 278, 286] [93, 122, 131, 267, 279, 287] [96, 124, 132, 200, 281, 288] [97, 125, 133, 271, 282, 289] [98, 126, 134, 272, 283, 290] [99, 127, 135, 273, 284, 291] [100, 128, 136, 274, 285, 292] [101, 129, 137, 275, 286, 293] [102, 130, 138, 276, 287, 294] [106, 132, 139, 210, 289, 295] [107, 133, 140, 280, 290, 296] [108, 134, 141, 281, 291, 297] [109, 135, 142, 282, 292, 298] [110, 136, 143, 283, 293, 299] [111, 137, 144, 284, 294, 300] [116, 139, 145, 220, 296, 301] [117, 140, 146, 288, 297, 302] [118, 141, 147, 289, 298, 303] [119, 142, 148, 290, 299, 304] [120, 143, 149, 291, 300, 305] [125, 145, 150, 230, 302, 306] [126, 146, 151, 295, 303, 307] [127, 147, 152, 296, 304, 308] [128, 148, 153, 297, 305, 309] [133, 150, 154, 240, 307, 310] [135, 152, 155, 302, 309, 311]
raw submission JSON