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[[234,8,18]] d ≤
n
234
k
8
d
18
kd²/n
11.077
w
6

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Distance

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