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[[318,4,26]] d ≤
n
318
k
4
d
26
kd²/n
8.503
w
6

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Distance

d_X 26 · witness weight 26 (claimed upper_bound)
witness operator (support, 26 qubits)
[5, 14, 17, 26, 31, 34, 40, 43, 49, 81, 87, 98, 107, 113, 122, 131, 145, 179, 183, 217, 237, 257, 294, 304, 309, 313]
d_Z 28 · witness weight 28 (claimed upper_bound)
witness operator (support, 28 qubits)
[7, 20, 22, 26, 55, 59, 67, 79, 85, 93, 101, 105, 109, 116, 121, 124, 137, 143, 146, 152, 159, 178, 185, 201, 210, 219, 254, 293]
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+x3y-4, g(x,y)=1+y+x-1y-3 on the abelian quotient group Z2/L with twist basis a_1=[0, 159], a_2=[1, 17] (n=2|det[a_1,a_2]|=2*159). 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 159 · Z-checks 159
H_X (159 checks, sparse supports)
[0, 1, 121, 159, 161, 257] [1, 3, 129, 160, 163, 265] [2, 4, 130, 161, 164, 266] [3, 6, 136, 162, 166, 273] [4, 7, 137, 163, 167, 274] [5, 8, 138, 164, 168, 275] [6, 10, 142, 165, 170, 281] [7, 11, 143, 166, 171, 282] [8, 12, 144, 167, 172, 283] [9, 13, 145, 168, 173, 284] [10, 15, 147, 169, 175, 289] [11, 16, 148, 170, 176, 290] [12, 17, 149, 171, 177, 291] [13, 18, 150, 159, 172, 178] [6, 14, 19, 173, 179, 292] [15, 21, 140, 174, 181, 296] [16, 22, 152, 175, 182, 297] [17, 23, 153, 176, 183, 298] [18, 24, 154, 160, 177, 184] [10, 19, 25, 161, 178, 185] [11, 20, 26, 179, 186, 299] [5, 21, 28, 180, 188, 302] [22, 29, 146, 181, 189, 303] [23, 30, 156, 182, 190, 304] [24, 31, 157, 162, 183, 191] [15, 25, 32, 163, 184, 192] [16, 26, 33, 164, 185, 193] [17, 27, 34, 186, 194, 305] [8, 28, 36, 187, 196, 307] [9, 29, 37, 188, 197, 308] [30, 38, 151, 189, 198, 309] [31, 39, 158, 165, 190, 199] [21, 32, 40, 166, 191, 200] [22, 33, 41, 167, 192, 201] [23, 34, 42, 168, 193, 202] [24, 35, 43, 194, 203, 310] [12, 27, 36, 195, 204, 311] [13, 37, 45, 196, 205, 312] [14, 38, 46, 197, 206, 313] [39, 47, 155, 169, 198, 207] [28, 40, 48, 170, 199, 208] [29, 41, 49, 171, 200, 209] [30, 42, 50, 172, 201, 210] [31, 43, 51, 173, 202, 211] [32, 44, 52, 203, 212, 314] [18, 35, 45, 204, 213, 315] [19, 46, 54, 205, 214, 316] [20, 47, 55, 174, 206, 215] [36, 48, 56, 175, 207, 216] [37, 49, 57, 176, 208, 217] [38, 50, 58, 177, 209, 218] [39, 51, 59, 178, 210, 219] [40, 52, 60, 179, 211, 220] [41, 53, 61, 195, 212, 221] [25, 44, 54, 213, 222, 317] [26, 55, 63, 180, 214, 223] [27, 56, 64, 181, 215, 224] [45, 57, 65, 182, 216, 225] [46, 58, 66, 183, 217, 226] [47, 59, 67, 184, 218, 227] [48, 60, 68, 185, 219, 228] [49, 61, 69, 186, 220, 229] [50, 62, 70, 204, 221, 230] [33, 53, 63, 187, 222, 231] [34, 64, 72, 188, 223, 232] [35, 65, 73, 189, 224, 233] [54, 66, 74, 190, 225, 234] [55, 67, 75, 191, 226, 235] [56, 68, 76, 192, 227, 236] [57, 69, 77, 193, 228, 237] [58, 70, 78, 194, 229, 238] [59, 71, 79, 213, 230, 239] [42, 62, 72, 196, 231, 240] [43, 73, 81, 197, 232, 241] [44, 74, 82, 198, 233, 242] [63, 75, 83, 199, 234, 243] [64, 76, 84, 200, 235, 244] [65, 77, 85, 201, 236, 245] [66, 78, 86, 202, 237, 246] [67, 79, 87, 203, 238, 247] [68, 80, 88, 222, 239, 248] [51, 71, 81, 205, 240, 249] [52, 82, 90, 206, 241, 250] [53, 83, 91, 207, 242, 251] [72, 84, 92, 208, 243, 252] [73, 85, 93, 209, 244, 253] [74, 86, 94, 210, 245, 254] [75, 87, 95, 211, 246, 255] [76, 88, 96, 212, 247, 256] [77, 89, 97, 231, 248, 257] [60, 80, 90, 214, 249, 258] [61, 91, 99, 215, 250, 259] [62, 92, 100, 216, 251, 260] [81, 93, 101, 217, 252, 261] [82, 94, 102, 218, 253, 262] [83, 95, 103, 219, 254, 263] [84, 96, 104, 220, 255, 264] [85, 97, 105, 221, 256, 265] [86, 98, 106, 240, 257, 266] [69, 89, 99, 223, 258, 267] [70, 100, 108, 224, 259, 268] [71, 101, 109, 225, 260, 269] [90, 102, 110, 226, 261, 270] [91, 103, 111, 227, 262, 271] [92, 104, 112, 228, 263, 272] [93, 105, 113, 229, 264, 273] [94, 106, 114, 230, 265, 274] [95, 107, 115, 249, 266, 275] [78, 98, 108, 232, 267, 276] [79, 109, 117, 233, 268, 277] [80, 110, 118, 234, 269, 278] [99, 111, 119, 235, 270, 279] [100, 112, 120, 236, 271, 280] [101, 113, 121, 237, 272, 281] [102, 114, 122, 238, 273, 282] [103, 115, 123, 239, 274, 283] [104, 116, 124, 258, 275, 284] [87, 107, 117, 241, 276, 285] [88, 118, 126, 242, 277, 286] [89, 119, 127, 243, 278, 287] [108, 120, 128, 244, 279, 288] [109, 121, 129, 245, 280, 289] [110, 122, 130, 246, 281, 290] [111, 123, 131, 247, 282, 291] [112, 124, 132, 159, 248, 283] [0, 113, 125, 267, 284, 292] [96, 116, 126, 250, 285, 293] [97, 127, 134, 251, 286, 294] [98, 128, 135, 252, 287, 295] [117, 129, 136, 253, 288, 296] [118, 130, 137, 254, 289, 297] [119, 131, 138, 255, 290, 298] [120, 132, 139, 160, 256, 291] [2, 122, 133, 276, 292, 299] [105, 125, 134, 259, 293, 300] [106, 135, 141, 260, 294, 301] [107, 136, 142, 261, 295, 302] [126, 137, 143, 262, 296, 303] [127, 138, 144, 263, 297, 304] [128, 139, 145, 162, 264, 298] [5, 131, 140, 285, 299, 305] [114, 133, 141, 268, 300, 306] [115, 142, 147, 269, 301, 307] [116, 143, 148, 270, 302, 308] [134, 144, 149, 271, 303, 309] [135, 145, 150, 165, 272, 304] [9, 139, 146, 293, 305, 310] [123, 140, 147, 277, 306, 311] [124, 148, 152, 278, 307, 312] [125, 149, 153, 279, 308, 313] [141, 150, 154, 169, 280, 309] [3, 14, 151, 300, 310, 314] [132, 146, 152, 286, 311, 315] [0, 153, 156, 287, 312, 316] [133, 154, 157, 174, 288, 313] [7, 20, 155, 195, 306, 314] [1, 151, 156, 294, 315, 317] [2, 157, 158, 180, 295, 316] [4, 155, 158, 187, 301, 317]
H_Z (159 checks, sparse supports)
[0, 13, 124, 159, 284, 312] [1, 18, 132, 159, 160, 315] [0, 2, 19, 161, 292, 316] [3, 24, 139, 160, 162, 310] [1, 4, 25, 161, 163, 317] [2, 5, 26, 164, 180, 299] [6, 31, 145, 162, 165, 173] [3, 7, 32, 163, 166, 314] [4, 8, 33, 164, 167, 187] [5, 9, 34, 168, 188, 305] [10, 39, 150, 165, 169, 178] [6, 11, 40, 166, 170, 179] [7, 12, 41, 167, 171, 195] [8, 13, 42, 168, 172, 196] [9, 14, 43, 173, 197, 310] [15, 47, 154, 169, 174, 184] [10, 16, 48, 170, 175, 185] [11, 17, 49, 171, 176, 186] [12, 18, 50, 172, 177, 204] [13, 19, 51, 173, 178, 205] [14, 20, 52, 179, 206, 314] [21, 55, 157, 174, 180, 191] [15, 22, 56, 175, 181, 192] [16, 23, 57, 176, 182, 193] [17, 24, 58, 177, 183, 194] [18, 25, 59, 178, 184, 213] [19, 26, 60, 179, 185, 214] [20, 27, 61, 186, 195, 215] [28, 63, 158, 180, 187, 199] [21, 29, 64, 181, 188, 200] [22, 30, 65, 182, 189, 201] [23, 31, 66, 183, 190, 202] [24, 32, 67, 184, 191, 203] [25, 33, 68, 185, 192, 222] [26, 34, 69, 186, 193, 223] [27, 35, 70, 194, 204, 224] [36, 53, 155, 187, 195, 207] [28, 37, 72, 188, 196, 208] [29, 38, 73, 189, 197, 209] [30, 39, 74, 190, 198, 210] [31, 40, 75, 191, 199, 211] [32, 41, 76, 192, 200, 212] [33, 42, 77, 193, 201, 231] [34, 43, 78, 194, 202, 232] [35, 44, 79, 203, 213, 233] [36, 45, 62, 196, 204, 216] [37, 46, 81, 197, 205, 217] [38, 47, 82, 198, 206, 218] [39, 48, 83, 199, 207, 219] [40, 49, 84, 200, 208, 220] [41, 50, 85, 201, 209, 221] [42, 51, 86, 202, 210, 240] [43, 52, 87, 203, 211, 241] [44, 53, 88, 212, 222, 242] [45, 54, 71, 205, 213, 225] [46, 55, 90, 206, 214, 226] [47, 56, 91, 207, 215, 227] [48, 57, 92, 208, 216, 228] [49, 58, 93, 209, 217, 229] [50, 59, 94, 210, 218, 230] [51, 60, 95, 211, 219, 249] [52, 61, 96, 212, 220, 250] [53, 62, 97, 221, 231, 251] [54, 63, 80, 214, 222, 234] [55, 64, 99, 215, 223, 235] [56, 65, 100, 216, 224, 236] [57, 66, 101, 217, 225, 237] [58, 67, 102, 218, 226, 238] [59, 68, 103, 219, 227, 239] [60, 69, 104, 220, 228, 258] [61, 70, 105, 221, 229, 259] [62, 71, 106, 230, 240, 260] [63, 72, 89, 223, 231, 243] [64, 73, 108, 224, 232, 244] [65, 74, 109, 225, 233, 245] [66, 75, 110, 226, 234, 246] [67, 76, 111, 227, 235, 247] [68, 77, 112, 228, 236, 248] [69, 78, 113, 229, 237, 267] [70, 79, 114, 230, 238, 268] [71, 80, 115, 239, 249, 269] [72, 81, 98, 232, 240, 252] [73, 82, 117, 233, 241, 253] [74, 83, 118, 234, 242, 254] [75, 84, 119, 235, 243, 255] [76, 85, 120, 236, 244, 256] [77, 86, 121, 237, 245, 257] [78, 87, 122, 238, 246, 276] [79, 88, 123, 239, 247, 277] [80, 89, 124, 248, 258, 278] [81, 90, 107, 241, 249, 261] [82, 91, 126, 242, 250, 262] [83, 92, 127, 243, 251, 263] [84, 93, 128, 244, 252, 264] [85, 94, 129, 245, 253, 265] [86, 95, 130, 246, 254, 266] [87, 96, 131, 247, 255, 285] [88, 97, 132, 248, 256, 286] [0, 89, 98, 257, 267, 287] [90, 99, 116, 250, 258, 270] [91, 100, 134, 251, 259, 271] [92, 101, 135, 252, 260, 272] [93, 102, 136, 253, 261, 273] [94, 103, 137, 254, 262, 274] [95, 104, 138, 255, 263, 275] [96, 105, 139, 256, 264, 293] [1, 97, 106, 257, 265, 294] [2, 98, 107, 266, 276, 295] [99, 108, 125, 259, 267, 279] [100, 109, 141, 260, 268, 280] [101, 110, 142, 261, 269, 281] [102, 111, 143, 262, 270, 282] [103, 112, 144, 263, 271, 283] [104, 113, 145, 264, 272, 284] [3, 105, 114, 265, 273, 300] [4, 106, 115, 266, 274, 301] [5, 107, 116, 275, 285, 302] [108, 117, 133, 268, 276, 288] [109, 118, 147, 269, 277, 289] [110, 119, 148, 270, 278, 290] [111, 120, 149, 271, 279, 291] [112, 121, 150, 159, 272, 280] [6, 113, 122, 273, 281, 292] [7, 114, 123, 274, 282, 306] [8, 115, 124, 275, 283, 307] [9, 116, 125, 284, 293, 308] [117, 126, 140, 277, 285, 296] [118, 127, 152, 278, 286, 297] [119, 128, 153, 279, 287, 298] [120, 129, 154, 160, 280, 288] [10, 121, 130, 161, 281, 289] [11, 122, 131, 282, 290, 299] [12, 123, 132, 283, 291, 311] [14, 125, 133, 292, 300, 313] [126, 134, 146, 286, 293, 303] [127, 135, 156, 287, 294, 304] [128, 136, 157, 162, 288, 295] [15, 129, 137, 163, 289, 296] [16, 130, 138, 164, 290, 297] [17, 131, 139, 291, 298, 305] [20, 133, 140, 174, 299, 306] [134, 141, 151, 294, 300, 309] [135, 142, 158, 165, 295, 301] [21, 136, 143, 166, 296, 302] [22, 137, 144, 167, 297, 303] [23, 138, 145, 168, 298, 304] [27, 140, 146, 181, 305, 311] [141, 147, 155, 169, 301, 306] [28, 142, 148, 170, 302, 307] [29, 143, 149, 171, 303, 308] [30, 144, 150, 172, 304, 309] [35, 146, 151, 189, 310, 315] [36, 147, 152, 175, 307, 311] [37, 148, 153, 176, 308, 312] [38, 149, 154, 177, 309, 313] [44, 151, 155, 198, 314, 317] [45, 152, 156, 182, 312, 315] [46, 153, 157, 183, 313, 316] [54, 156, 158, 190, 316, 317]
raw submission JSON