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[[248,10,18]] d ≤
n
248
k
10
d
18
kd²/n
13.065
w
6

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Distance

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