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[[270,8,20]] d ≤
n
270
k
8
d
20
kd²/n
11.852
w
6

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Distance

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